MP02: Motion DiagramsVelocity and Acceleration of a Power Ball Learning Goal: To understand the distinction between velocity and acceleration with the use of motion diagrams. In common usage, velocity and acceleration both can imply having considerable speed. In physics, they are sharply defined concepts that are not at all synonymous. Distinguishing clearly between them is a prerequisite to understanding motion. Moreover, an easy way to study motion is to draw a motion diagram, in which the position of the object in motion is sketched at several equally spaced instants of time, and these sketches (or snapshots) are combined into one single picture. In this problem, we make use of these concepts to study the motion of a power ball. This discussion assumes that we have already agreed on a coordinate system from which to measure the position of objects as a function of time. Let and (also called the position vector) be velocity and acceleration, respectively. Harvaran Ghai Consider the motion of a power ball that is dropped on the floor and bounces back. In the following questions, you will describe its motion at various points in its fall in terms of its velocity and acceleration. Part A You drop a power ball on the floor. The motion diagram of the ball is sketched in the figure the magnitude of the velocity of the ball is increasing, decreasing, or not changing. increasing . Indicate whether decreasing not changing Correct While the ball is in free fall, the magnitude of its velocity is increasing, so the ball is accelerating. Part B Since the length of is directly proportional to the length of , the vector connecting each dot to the next could represent velocity vectors as well as position vectors, as shown in the figure here . Indicate whether the velocity and acceleration of the ball are, respectively, positive (upward), negative, or zero. Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma. N,N Correct Part C Now, consider the motion of the power ball once it bounces upward. Its motion diagram is shown in the figure here . Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing. increasing decreasing not changing Correct Since the magnitude of the velocity of the ball is decreasing, the ball must be accelerating (specifically, slowing down). Part D The next figure shows the velocity vectors corresponding to the upward motion of the power ball. Indicate whether its velocity and acceleration, respectively, are positive (upward), negative, or zero. Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma. P,N Correct Part E The power ball has now reached its highest point above the ground and starts to descend again. The motion diagram representing the velocity vectors is the same as that after the initial release, as shown in the figure of Part B. Indicate whether the velocity and acceleration of the ball at its highest point are positive (upward), negative, or zero. Use P, N, and Z for positive (upward), negative, and zero, respectively. Separate the letters for velocity and acceleration with a comma. Z,N Correct These examples should show you that the velocity and acceleration can have opposite or similar signs or that one of them can be zero while the other has either sign. Try hard to think carefully about them as distinct physical quantities when working with kinematics. Motion of Two Rockets Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a timeexposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated. Harvaran Ghai Part A At what time(s) do the rockets have the same velocity? at time only at time only at times and at some instant in time between and at no time shown in the figure Correct Part B At what time(s) do the rockets have the same x position? at time only at time only at times and at some instant in time between and and nonzero acceleration velocity displacement time Correct Part F At what time(s) is rocket A ahead of rocket B? before after before only only and after .e. constant) __________.at no time shown in the figure Correct Part C At what time(s) do the two rockets have the same acceleration? at time only at time only at times and at some instant in time between and at no time shown in the figure Correct Part D The motion of the rocket labeled A is an example of motion with uniform (i. constant) __________.. and nonzero acceleration velocity displacement time Correct Part E The motion of the rocket labeled B is an example of motion with uniform (i..e. Harvaran Ghai Represent the moving object as a particle. Use five or six dots to make the motion clear but without overcrowding the picture. A car is traveling with constant velocity along a highway. Make simplifying assumptions when interpreting the problem statement. you will be asked several questions related to construction of a motion diagram for this situation and a few others. The car has just achieved double its initial velocity when the driver spots a policeman behind him and applies the brakes.between and at no time(s) shown in the figure Correct PSS 1. . coming to rest at a stoplight ahead. MODEL: VISUALIZE: A complete motion diagram consists of: The position of the object in each frame of the film.1: (Almost) a Dozen Diagrams Learning Goal: To practice ProblemSolving Strategy 1. More complex motions may need more dots. The car then decelerates. The driver notices he is late for work so he stomps down on the gas pedal and the car begins to accelerate. shown as a dot.1 for constructing motion diagrams. In this problem. e. if you think that assumptions C and D are reasonable. Enter the letters of all the correct answers in alphabetical order.. including all the elements listed in the problemsolving strategy. there are no curves). There is one velocity vector linking each set of two position dots. Part B Which of the diagrams best describes the position and the velocity of the car before the driver notices he is late? A B C Correct . The highway is level (i. C. B. The highway is straight (i. D.e.3. Refer to this set of motion diagrams in answering the following. During each of the three different stages of its motion. found using Tactics Box 1. Each acceleration vector is drawn at the dot between the two velocity vectors it links. the car is moving with constant (possibly zero) velocity. Part A Which of the following simplifying assumptions is it reasonable to make in this problem? A. enter CD. assume that the car is moving in a straight line to the right. Label the row of acceleration vectors . found by connecting each dot in the motion diagram to the next with a vector arrow. During each of the three different stages of its motion.. the car is moving with constant (possibly zero) acceleration. In interpreting the diagrams that follow. there are no hills or valleys). ACD Correct Visualize Now draw a motion diagram. The average velocity vectors. The average acceleration vectors. Model It is appropriate to use the particle model for the car. You should also make some simplifying assumptions. There is one acceleration vector linking each set of two velocity vectors. Use to indicate a point at which the acceleration is zero. Use your diagram to answer the following questions. Do not use commas. For example. Label the row of velocity vectors . which object corresponds to the position and velocity diagram shown here? the train the sled . the sled comes to a stop after covering some distance. A B C Correct Now let's use our results for the car and apply them to some other problems. but before he notices the policeman? A B C Correct Part D Which of the diagrams best describes the position and the velocity of the car after the driver notices the policeman? A B C Correct Part E Which of these diagrams problem introduction? most accurately depicts the acceleration of the car during the events described in the Assume that the car is initially moving to the right. Part F Of the three situations described. A sled is given a quick push along a horizontal surface. Consider these three situations: A train has its brakes released and pulls out of the station. slowly picking up speed. A motorcycle is moving along a straight highway at 105 km/h (the legal speed limit in many states).Part C Which of the diagrams best describes the position and the velocity of the car after the driver hits the gas. Assume that both cars are moving to the right. The car and the truck have the same acceleration. The car takes less time to stop than the truck. Both drivers apply the brakes at the same moment. The car is moving at constant velocity. which object corresponds to the motion diagram shown here? the train the sled the motorcycle Correct Let us now consider another scenario. Part H Which of the three diagrams shown best describes the motion of the car and truck after the brakes have been applied? A B C Correct Part I Diagram (B) is incorrect because. according to it: The car and truck move in different directions. A car and a truck are moving at the same velocity along a straight highway. The truck is speeding up. it lacks the vector representing the acceleration of the object. Correct .the motorcycle Correct Note that the diagram shown is not a complete motion diagram. Refer to the motion diagrams shown here in answering the following. The car and truck both come to a stop. Part G Of the three situations described. Correct MP04: Using Motion Diagrams Curved Motion Diagram The motion diagram shown here represents a pendulum released from rest at an angle of 45 from the vertical. The car is moving at constant velocity. The car and the truck have the same acceleration. The dots in the motion diagram represent the positions of the pendulum bob at eleven moments separated by equal time intervals. Harvaran Ghai Part A What is the direction of the acceleration of the pendulum bob at moment 5? Enter the letter of the arrow with this direction from the compass rose in the figure. The truck is speeding up. Type Z if the acceleration vector has zero length. Also given is a "compass rose" of directions in which the different directions are labeled with the letters of the alphabet. according to it: The car and truck move in different directions. The green arrows represent the average velocity between adjacent dots.Part J Diagram (C) is incorrect because. A Correct . Based on observation or comparison with other reallife pendula. A race car rounding a turn D. A marble released part way up the inside surface of a smoothly rounded bowl Enter the letters of all possible correct scenarios in alphabetical order. A factor of 3 error is allowed in either direction. AD Correct Part D Assume that the diagram in the problem introduction represents the motion of a ball tied on the end of a stringthat is.Part B What is the direction of the acceleration of the pendulum bob at moments 0 and 10? Enter the letters of the arrows with these directions from the compass rose in the figure. is then approximately 2 s.0 Correct Physics can often seem to be a science of very precise answers. Do not use commas. =1. having a solid grasp of the fundamental concepts allows a physicist to make reasonably accurate estimates like this one very quickly. an initial estimate gives you a way of checking whether the result of a long calculation is reasonable. also called the period of the pendulum. An interesting fact about the pendulum is that its period is essentially independent of the weight of the ball (or whatever other object is used). directions at moment 0. moment 10 =D. . A weight placed on the rim of a bicycle wheel that is being held off the ground so it can rotate freely B. The total time it takes for this pendulum to swing back and forth. Express in meters. However.F Correct Part C In which of the following other scenarios could the motion reasonably be represented by the motion diagram in the introduction? A.10 s. a pendulum. Even if you were ultimately looking for a more precise answer. Also assume that the interval between each time step in the diagram is 0. Type Z if the acceleration vector has zero length. estimate the length of the string needed for the pendulum to have a period of 2 s. separated by commas. It depends only on the length of the string (with longer period for longer strings) and the strength of the force of gravity (which is essentially constant over the surface of the earth). An airplane pulling out of a dive C. =20 Correct Part C Now that you have and . In this problem you must find the average velocity from a graph of indicate the average velocity over the time interval from to .Average Velocity from a Position vs. Answer to the nearest integer. Thus interval from 1 to 3 s. . . We will use the notation to is the average velocity over the time Harvaran Ghai Part A Find the average velocity over the time interval from 0 to 1 second. Answer to the nearest integer. =0 Correct Part B Find the average velocity over the time interval from 1 to 3 seconds. find . Time Graph Learning Goal: To learn to read and interpolate on a graph of position versus time and to change units. Round your answer to two significant figures. because they are averages because it is for an Part D Find the average velocity over the time interval from 1 to 5 seconds. with . Tim walks half of the time and runs the other Harvaran Ghai Part A . Do not simply eyeball the position or you will likely not be able to obtain the solution to the desired accuracy. Express your answer to two significant figures. Correct =8.5 to 6. =6. You would have to double the weight given to interval twice as long. Express your answer to the nearest integer. .6 Running and Walking Tim and Rick both can run at speed and walk at speed distance half. =13.7 Correct Part E Obtaining this answer required some interpolation on the graph.3 Correct Note that is not equal to the simple arithmetic average of and for time intervals of different length. They set off together on a journey of .Give your answer to three significant figures. You will need to interpolate to find the position at time .0 seconds. =24 Correct Part F Find the average velocity over the time interval from 2. Rick walks half of the distance and runs the other half. Now see if you can express this result in terms of kilometers per hour. . = Correct . . Correct Part E In terms of given quantities. Part B Find Rick's average speed for covering the distance Express Rick's average speed in terms of . Rick Tim Neither. . . They cover the distance in the same amount of time. and . but without using the detailed answers in the previous parts. and . Part D Who covers the distance more quickly? Think logically. . . does Tim beat Rick? It will help you check your answer if you simplify it algebraically and check the special case Express the difference in time. = Correct . by what amount of time.How long does it take Rick to cover the distance ? Express the time taken by Rick in terms of . and . and . Part C How long does it take Tim to cover the distance? Express the time taken by Tim in terms of . = Correct . = Correct Part F in terms of . to to to to Correct to . =55 Correct Part B During which time interval is the acceleration positive? Indicate the most complete answer. Use the graph to answer the following Harvaran Ghai Part A Find the maximum velocity of the car. Express your answer in meters per second to the nearest integer.In the special case that Correct . shows the velocity of a sports car as a function of time . what would be Tim's margin of victory ? 0 Graph of v(t) for a Sports Car The graph questions. Express your answer in meters per second squared to the nearest integer. Express your answer in meters per second squared to the nearest integer. Let the time at which the dragster starts to accelerate be .) You drive at a constant speed of toward the stopped dragster. The dragster driver sees you coming but waits until the last instant to put down the hammer. =30 Correct Part D Find the minimum magnitude of the acceleration of the car. =0 Correct Part E Find the distance traveled by the car between 0 and 2 s. (Burning out means spinning the tires at high speed to heat the tread and make the rubber sticky. you attempt to run your car into the back of a dragster that is "burning out" at the red light before the start of a race. =55 Correct Rearending Drag Racer To demonstrate the tremendous acceleration of a top fuel drag racer. not slowing down in the face of the imminent collision. Express your answer in meters to the nearest integer. accelerating from the starting line at constant acceleration.Part C Find the maximum acceleration of the car. . . and give your answer to two significant figures.7.2 s. choose coordinates so that the position of the drag car is 0 at . = Correct Part C Find numerical values for and in seconds and meters for the (reasonable) values and . If you calculate positions on the way to this solution.8 m/s) . the longest time after the dragster begins to accelerate that you can possibly run into the back of the dragster if you continue at your initial velocity? = Correct Part B Assuming that the dragster has started at the last instant possible (so your front bumper almost hits the rear of the dragster at ). and can never be negative). .54.Harvaran Ghai Part A What is . m (26. Correct =0. Separate your two numerical answers by commas. find your distance from the dragster when he started. not a position (which can be negative). Remember that you are solving for a distance (which is a magnitude. the object is located at the source . . Motion of a Shadow A small source of light is located at a distance from a vertical wall. . continues at constant velocity (blue) and just barely touches the . the magnitude of the velocity of the top of the object's shadow. at time . Express the speed of the top of the object's shadow in terms of . . At time . and . An opaque object with a height of moves toward the wall with constant velocity of magnitude . Part A Harvaran Ghai Find an expression for .The blue curve shows how the car. initially at accelerating drag car (red) at . 5(t1)2*a) Correct Part B If the rocket's net acceleration is for Express your answer numerically in meters. Express the maximum height in terms of . and/or . so that downward velocities are positive and the acceleration due to gravity is the positive quantity . Harvaran Ghai Part A Find the maximum height that the rocket reaches (neglecting air resistance). when the fuel is exhausted. Assume that the flower pot was dropped by someone on the floor above you (rather than thrown downward). =1470 m Correct A Flower Pot Falling Past a Window As you look out of your dorm window. what is the maximum height the rocket will reach? . accelerates straight upward from rest with constant net acceleration .5g((a*t1)/g)2)+(. Part A From what height above the bottom of your window was the flower pot dropped? Harvaran Ghai . = a*t1*((a*t1)/g)(. The pot is visible for a time . until time . = Correct Rocket Height A rocket. initially at rest on the ground. a flower pot suddenly falls past. is a positive number equal to the magnitude of the acceleration due to gravity. and the vertical length of your window is . Note that in this problem. . using . Take down to be the positive direction. a rope with tension exerts a force of magnitude in a direction 35 degrees north of east. This is a good way to think of vectors. Express your answer in terms of some or all of the variables = Correct . defined by . and .Express your answer in terms of . = Correct Part B If the bottom of your window is a height above the ground. . . . to calculate results with vectors. the speed at the bottom of the window. . and . it is best to select a coordinate system and manipulate the components of the vectors in that coordinate system. for example. however. Resolving Vector Components with Trigonometry Often a vector is specified by a magnitude and a direction. . what is the velocity of the pot as it hits the ground? You may introduce the new variable . Don't forget that when multiplying two factors. = Correct . Write the components in the form x. the cos and sin functions must have parentheses around their arguments.y. you must include a multiplication symbol. = Correct Notice that vectors and have the same form despite their placement with respect to the y axis on the drawing. = Correct Part B Find the components of the vector with length and angle with respect to the x axis as shown.Harvaran Ghai Part A Find the components of the vector with length and angle with respect to the x axis as shown.m*cos(N). a vector might take the form p*sin(Q). named . also. Part C Find the components of the vector with length and angle as shown. Write the components in the form x. For example. named . Express your answer in terms of and .y. named . Write the components in the form x.y. The above the horizon.26 Correct .Tracking a Plane A radar station. What are the ? Express in meters as an ordered pair. =1100. detects an airplane coming straight at the station from the east. where the x axis represents the ground and the positive z direction is upward. The contact points are shown in the diagram. The position of point B relative to the origin is (the magnitude of is 880 ). to two significant figures. located at the origin of xz plane. separating the x and z components with a comma. as shown in the figure . the position of the airplane relative to the origin is position vector has a magnitude of 360 tracked for another 123 and is located at exactly 40 . until it has passed directly over the station and reached point B. The airplane is in the vertical eastwest plane for 5.0 . Harvaran Ghai Part A Define the displacement of the airplane while the radar was tracking it: components of . At first observation (point A). 0 above the has a magnitude of 6. negative x axis in the second quadrant. Although such a definition may not sound too scientific. . Each force acts upon some other object. A force can be simply defined as a push or a pull exerted by one object upon another. act at a point. has a magnitude of 9. and .36 Correct Part C What is the magnitude of the resultant force? Express your answer in newtons. it does capture three essential properties of forces: Each force is created by some object. 9.2 below the Harvaran Ghai Part A What is the x component of the resultant force? Express your answer in newtons. negative x axis in the third quadrant.73 Correct Part B What is the y component of the resultant force? Express your answer in newtons. 8. 3.80 and is directed at an angle of 60.40 and is directed at an angle of 53.Two Forces Acting at a Point Two forces.36 Correct A Push or a Pull? Learning Goal: To understand the concept of force as a push or a pull and to become familiar with everyday forces. consider a book resting on a horizontal table. In this problem. Note that such a distinction is useful but not really fundamental: For instance. you will identify the types of forces acting on objects in various situations." One of the biggest mistakes you may make is to think of a force as "something an object has." In fact. friction. the force of tension. It is sometimes convenient to classify forces as either contact forces between two objects that are touching or as longrange forces between two objects that are some distance apart. at least two objects are always required for a force to exist." "Object A exerts force upon object B. The action of a force can be visualized as a push or a pull. Part A Which object exerts a downward force on the book? the book itself the earth the surface of the table Correct . and the normal force. Each force has a direction: Forces are vectors. The proper words describing the force interaction between objects A and B may be any of the following: "Object A acts upon object B with force . forces must be described as interactions. Contact forces include tension. on a microscopic scale the force of friction is really an electromagnetic force. The main result of such interactions is that the objects involved change their velocities: Forces cause acceleration. Longrange forces include gravity and electromagnetic forces. in this problem. However." "Force due to object A is acting upon object B. First. Since each force is created by one object and acts upon another. the force of friction. and the normal force. we will not concern ourselves with accelerationnot yet. Harvaran Ghai Some common types of forces that you will be dealing with include the gravitational force (weight)." "Force is applied to object B by object A. Part B The downward force acting on the book is __________. a contact force a longrange force Correct Part F What is the upward force acting on the book called? tension normal force weight friction . a contact force a longrange force Correct Part C What is the downward force acting on the book called? tension normal force weight friction Correct Part D Which object exerts an upward force on the book? the book itself the earth the surface of the table Correct Part E The upward force acting on the book is __________. the string must be connected to (i.e. touching) the block. a contact force a longrange force Correct To exert a tension force. A string is attached to a heavy block. Part G Which object exerts a force on the block that is directed toward the right? the block itself the earth the surface of the table the string Correct Part H The force acting on the block and directed to the right is __________. Part I What is the force acting on the block and directed to the right called? tension normal force weight friction Correct Part J Which object exerts a force on the block that is directed toward the left? the block itself the earth the surface of the table the string Correct .Correct Now consider a different situation.. The string is used to pull the block to the right along a rough horizontal table. The following questions refer to the motion of the block after it is pushed but before it stops. the normal force exerted by a student's hand or some spring launcher). The block slides to the right and eventually stops. Part M How many forces are acting on the block in the horizontal direction? 0 1 2 3 Correct Once the push has commenced.Part K The force acting on the block and directed to the left is __________. The same block is placed on the same rough table. Once the contact with the launching object has been lost. Part N What is the force acting on the block that is directed to the left called? tension normal force . a contact force a longrange force Correct Part L What is the force acting on the block and directed to the left called? tension normal force weight friction Correct Now consider a slightly different situation. this time. However. the only horizontal force acting on the block is directed to the leftwhich is why the block eventually stops. the string is disconnected and the block is given a quick push to the right. there is no force acting to the right: The block is moving to the right because it was given a velocity in this direction by some force that is no longer applied to the block (probably. " Once all of the forces are drawn. To find the net force. you must first identify all of the forces involved and then add them as vectors. Here is the general strategy for drawing freebody diagrams: Identify the object of interest. in turn. This may not always be easy: A sketch of the situation may contain many objects. There are two possible difficulties here: omitting some forces and drawing the forces that either don't exist at all or are applied to other objects. "every force must have a source. fricion becomes zero (assuming the table is perfectly horizontal). The forces should be shown as vectors originating from the dot representing the object of interest. That is. Once the block stops. The origin should coincide with the dot representing the object of interest and the axes should be chosen so that the subsequent calculations of vector components of the forces will be relatively simple. that analysis would involve finding the acceleration of the objects. Draw and clearly label all the forces acting on the object of interest. Draw the object as a dot. remember that every force must be applied to the object of interest by some other objector. To avoid these two pitfalls. Such a procedure is not always trivial. Frequently.weight friction Correct The force of friction does not disappear as long as the block is moving. This problem will walk you through several examples of freebody diagrams and will demonstrate some of the possible pitfalls. Including forces acting on different objects in the same diagram will lead to confusion and a wrong solution. . requires that you find the net force. Such a drawing is called a freebody diagram. It is helpful to replace the sketch of the situation by the drawing of the object (represented as a particle) and all the forces applied to it. as many forces as possible must be either parallel or perpendicular to one of the axes. each of which has a different set of forces acting on it. draw the coordinate system. Free-Body Diagrams: Introduction Learning Goal: To learn to draw freebody diagrams for various reallife situations. as some like to say. That. Imagine that you are given a description of a reallife situation and are asked to analyze the motion of the objects involved. The normal force exists between two surfaces that are pressed against each other. Keep in mind that to simplify problems you often assume friction is negligible on smooth surfaces. may come into play.g. Keep in mind that a spring can be either compressed or stretched whereas a string can only be stretched. force of velocity D. diagram for the puck. Force of tension. Here are the principal ones of interest: Weight. We will start with relatively simple situations in which the object of interest is either explicitly suggested or fairly obvious. These forces are directed so that they resist the relative motion of the surfaces. even though real life can present us with a wide variety of situations. normal force Draw a freebody . springs. force of push E. In addition. friction C. and other objects of finite length. smooth icy surface at a constant velocity as shown. Force of friction. the force of air drag. or the force due to gravity. Which of the following forces are acting on the puck? A. we will be mostly dealing with a very small number of forces. satellites). A friction force exists between two surfaces that either move or have a tendency to move relative to each other. it is always perpendicular to the surfaces. Sometimes.. Tension exists in strings. the word friction commonly refers to resistive forces other than air drag that are caused by contact between surfaces so you can ignore air drag in problems unless you are told to consider its effects. Weight acts on every object and is directed straight down unless we are considering a problem involving the nonflat earth (e. The following examples should help you learn to draw freebody diagrams.Harvaran Ghai It should come as good news that. weight B. It is directed along the string or a spring. Part A A hockey puck slides along a horizontal. Normal force. similar in some ways to the force of friction. force of velocity D. in the situation described." If the puck is not being pushed. For instance. Finally. type CD. normal force F. friction C. acceleration Type the letters corresponding to all the correct answers in alphabetical order. Draw a freebody diagram for the block. the word "smooth" usually implies negligible surface friction. Your freebody diagram should look like the one shown here. Which of the following forces are acting on the block? A. no such "force of push" exists. if you think that only answers C and D are correct. Do not use commas. weight B. air drag G. The tension in the rope is nonzero. there are no horizontal forces acting on it.F. force of tension E. type CD. AE Correct There is no such thing as "the force of velocity. Also. Do not use commas. air drag G. some horizontal force must have acted on it before. acceleration Type the letters corresponding to all the correct answers in alphabetical order. . For instance. if you think that only answers C and D are correct. Part B Consider a block pulled by a horizontal rope along a horizontal surface at a constant velocity as shown. to impart the velocityhowever. the air drag in such cases is assumed to be negligible. Of course. Your freebody diagram should look like that shown here. Do not use commas.ABDE Correct Because the velocity is constant. A block is resting on a slope as shown. if you think that only answers C and D are correct. force of push E. Part C Which of the following forces are acting on the block? A. weight B. For instance. type CD. Consider the following situation in parts C F. static friction D. ACE Correct Part D What is the direction of the force due to gravity acting on the block? vertically upward vertically downward perpendicular to the slope . kinetic friction C. it is kinetic friction. normal force Type the letters corresponding to all the correct answers in alphabetical order. there must be a force of friction opposing the force of tension. Since the block is moving. so the force of static friction must oppose such a motion and be directed upward along the slope. What is the direction of the force of friction acting on the block? vertically upward vertically downward perpendicular to the slope upward along the slope downward along the slope Correct Without friction. Your freebody diagram should look like that shown here.upward along the slope downward along the slope Correct Part E What is the direction of the normal force acting on the block? vertically upward vertically downward perpendicular to the slope upward along the slope downward along the slope Correct Part F Draw the freebody diagram for the block. the block would slide down the slope. . Once again. type CD. force of push E. Do not use commas. static friction D. if you think that only answers C and D are correct. weight B. the force of velocity Type the letters corresponding to all the correct answers in alphabetical order. For instance. Part G Which of the following forces are acting on the block? A. it is kinetic friction. kinetic friction C.Now consider a block sliding up a rough slope after having been given a quick push as shown. it seems a tempting choice to some students since the block is going up. ABE Correct The word "rough" implies the presence of friction. there is no such thing as "the force of velocity. What is the direction of the force of friction acting on the block? vertically upward vertically downward perpendicular to the slope upward along the slope downward along the slope Correct ." However. Part H Draw the freebody diagram for the block. normal force F. Since the block is in motion. The force pushing the block is parallel to the slope. ADE Correct . kinetic friction C. Your freebody diagram should look like the one shown here. type CD. Do not use commas. static friction D.The force of kinetic friction opposes the upward motion of the block. Part I Now consider a block being pushed up a smooth slope. Which of the following forces are acting on the block? A. normal force Type the letters corresponding to all the correct answers in alphabetical order. force of push E. if you think that only answers C and D are correct. For instance. weight B. the table F.Your freebody diagram should look like the one shown here. the block of mass B. Let us consider a situation where choosing the objects for which to draw the freebody diagrams is up to you. For instance. AB Correct Part K Draw the freebody diagram for the block of mass none one two . The block of mass is sliding to the right on a rough horizontal surface of a lab table. if you think that only answers C and D are correct. you will have to draw the freebody diagrams for which objects? A. by the palm of the hand of the person pushing the block. the block of mass C. In all the previous situations just described. Do not use commas. The force of push is the normal force exerted. the earth Type the letters corresponding to all the correct answers in alphabetical order. the connecting string D. Part J To solve for the acceleration of the blocks. possibly. the pulley E. type CD. Two blocks of masses and are connected by a light string that goes over a light frictionless pulley. the object of interest was explicitly given. How many forces are exerted on this block? . three four Correct Your freebody diagram should look like that shown here. How many forces are exerted on this block? . Part L Draw the freebody diagram for the block of mass none one two three four Correct . Understanding Newton's Laws Harvaran Ghai Part A An object cannot remain at rest unless which of the following holds? The net force acting on it is zero. the body accelerates. what can one conclude? There is exactly one force applied to the block. The net force could be zero either because there are no forces acting on the body at all or because several forces are acting on the body but they all cancel out. There is only one force acting on it. There must be no forces at all applied to the block. The net force applied to the block is directed to the left.Your freebody diagram should look like that shown here. regardless of whether it is a constant force. There are no forces at all acting on it. Correct . Part B If a block is moving to the left at a constant velocity. Correct If there is a net force acting on a body. The net force acting on it is constant and nonzero. The net force applied to the block is zero. then it will remain at rest. If the body is at rest and the net force acting on it is zero. the body does not move at a constant velocity. In this case. If a body is moving with constant velocity. The block must be __________. Part E Two forces. It could be moving to the left. of magnitude and . cannot be directed the same way as the force of . cannot be equal to B. (directed to the left) and (directed to the right). continuously changing direction moving at constant velocity moving with a constant nonzero acceleration moving with continuously increasing acceleration Correct Since there is a net force acting. Part C A block of mass is acted upon by two forces: you say about the block's motion? It must be moving to the left. the net force on (and therefore the acceleration of) the block is to the right. or be instantaneously at rest. or in any other direction. A. The relative direction of the forces is unknown. but the block could be moving left. The net force could be zero either because there are no forces acting on the body at all or because several forces are acting on the body but they all cancel out. are applied to an object. The net force acting on the object __________. moving to the right. regardless of whether the body is already moving. Hence. cannot be equal to C. However.If there is a net force acting on a body. Part D A massive block is being pulled along a horizontal frictionless surface by a constant horizontal force. the acceleration of the body is also constant. but it accelerates instead. the body accelerates. It must be at rest. then it is not accelerating and the net force acting on it is zero. according to Newton's 2nd law of motion. right. the force acting on the body is constant. Correct The acceleration of an object tells you nothing about its velocitythe direction and speed at which it is moving. What can It must be moving to the right. A Correct Conceptual Questions on Newton's 1st and 2nd Laws Learning Goal: To understand the meaning and the basic applications of Newton's 1st and 2nd laws. In this problem. must be greater than Enter the letters of all the correct answers in alphabetical order. The dots are connected by arrows representing the object's average velocity during the corresponding time interval. you are given a diagram representing the motion of an objecta motion diagram. enter D. Your goal is to use this motion diagram to determine the direction of the net force acting on the object.D. You will then determine which force diagrams and which situations may correspond to such a motion. . Do not use commas. Harvaran Ghai Part A What is the direction of the net force acting on the object at position A? upward downward to the left to the right The net force is zero. For example. if you think only the last option is correct. The dots represent the object's position at moments separated by equal intervals of time. however. Correct The horizontal component of the velocity does not change. the numbers corresponding to the correct diagrams. Part B What is the direction of the net force acting on the object at position B? upward downward to the left to the right The net force is zero. The number next to each arrow represents the magnitude of the force in newtons. These diagrams represent the forces acting on a moving object. The next four questions are related to the force diagrams numbered 1 to 6. the accelerationand the net forceare directed straight downward. the acceleration is directed to the leftand so is the net force. if you think that only diagrams 3 and 4 are correct. Do not use commas. Part D Which of these diagrams may possibly correspond to the situation at point A on the motion diagram? Type. it is decreasing. Therefore. in increasing order. type 34.Correct The velocity vectors connecting position A to the adjacent positions appear to have the same magnitude and direction. 6 Correct Part E . Part C What is the direction of the net force acting on the object at position C? upward downward to the left to the right The net force is zero. Therefore. For instance. The vertical component of the velocity increases. Therefore. Correct The velocity is directed to the right. the acceleration is zeroand so is the net force. H. the numbers corresponding to the correct diagrams. A rock is dropped vertically.e. type CD. air resistance is negligible. AD Correct . type 34. if you think that only diagrams 3 and 4 are correct. A hockey puck slides along a rough concrete surface. A car is moving along a straight road at a constant speed. A rock is thrown horizontally. Part H Which of these situations describe the motion shown in the motion diagram at point A? Type the letters corresponding to all the right answers in alphabetical order. D. J. For instance. I. Do not use commas. B. in increasing order. air resistance is negligible. if you think that only situations C and D are correct. For instance. A car is moving along a straight road while slowing down. 24 Correct Part G Which of these diagrams correspond to a situation where the moving object (not necessarily the one shown in the motion diagram) is changing its velocity? Type. in increasing order. A hockey puck slides along a smooth (i. G. air resistance is substantial. the numbers corresponding to the correct diagrams. A car is moving along a straight road while speeding up. 35 Correct Part F Which of these diagrams may possibly correspond to the situation at point C on the motion diagram? Type. A rock is dropped vertically. type 34.. A rock is thrown horizontally. in increasing order. if you think that only diagrams 3 and 4 are correct. if you think that only diagrams 3 and 4 are correct. C. F. A cockroach is speeding up from rest. Do not use commas. 12345 Correct Consider the following situations: A. For instance. Do not use commas.Which of these diagrams may possibly correspond to the situation at point B on the motion diagram? Type. Do not use commas. air resistance is substantial. frictionless) icy surface. E. type 34. the numbers corresponding to the correct diagrams. For instance. Do not use commas. Do not use commas. G Correct A World-Class Sprinter Worldclass sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude . sprinters push backward on the starting blocks with their feet. As a reaction. For instance. type CD. exert on the starting blocks to produce this =810 Correct Part B Which body exerts the force that propels the sprinter. the blocks push forward on their feet with a force of the same magnitude. Harvaran Ghai Part A How much horizontal force must a sprinter of mass 54. This external force accelerates the sprinter forward.0 acceleration? Express your answer in newtons. BE Correct Part J Which of these situations describe the motion shown in the motion diagram at point C? Type the letters corresponding to all the right answers in alphabetical order. . type CD. For instance.Part I Which of these situations describe the motion shown in the motion diagram at point B? Type the letters corresponding to all the right answersin alphabetical order. the blocks or the sprinter? the blocks the sprinter Correct To start moving forward. if you think that only situations C and D are correct. if you think that only situations C and D are correct. in which the y axis is vertical? Harvaran Ghai . with the x axis along the plane. the force of friction. The coefficient of friction is large enough to prevent the Part A Consider coordinate system a.Block on an Incline A block lies on a plane raised an angle from the horizontal. the force of and Part B Which forces lie along the axes of the coordinate system b. and block from sliding . . Which forces lie along the axes? only only only and and and and Correct . Three forces act upon the block: gravity. the normal force. . . .only only only and and and and Correct and Now you are going to ignore the general rule (actually. find an expression for = Correct since involving is an unknown. and . using vertical coordinate system b. especially unknown ones. Express your answer in terms of some or all of the variables . and . . a strong suggestion) that you should pick the coordinate system with the most vectors. using coordinate system b. Correct Part D Because the block is not moving. the sum of the y components of the forces acting on the block must be zero. . the sum of the x components of the forces acting on the block must be zero. Correct Part E To find the magnitude of the normal force. . Find an expression for the sum of the x components of the forces acting on the block. using coordinate system b. you must express in terms of equations you found in the two previous parts. Using the and but not . You will find the normal force. along the coordinate axes. each multiplied by a trigonometric function. Find an expression for the sum of the y components of the forces acting on the block. Express your answer in terms of some or all of the variables . In these coordinates you will find the magnitude and y equations. appearing in both the x Part C Because the block is not moving. . . Now realize that in coordinate system a.Congratulations on working this through. Because the ceiling is covered with intricate architectural decorations (not indicated in the figure. the ycoordinate equation is for . as well as the magnitude of the . which uses a humbler depiction). Hanging Chandelier A chandelier with mass is attached to the ceiling of a large concert hall by two cables. and makes an angle of with the ceiling. which is aligned with the plane. Cable 1 has tension Cable 2 has tension and makes an angle of with the ceiling. CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Instead. . Harvaran Ghai Part A Find an expression for . that does not depend on Express your answer in terms of some or all of the variables acceleration due to gravity . and . = Correct . they attached the cables to the ceiling near the walls. the workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier. the tension in cable 1. which leads immediately to the result obtained here . What is his weight on Mars. Harvaran Ghai Part A What is her weight on earth? 539 N Correct Part B What is her mass on the moon.12 A woman has a mass of . where 309 N Correct Problem 5. where 55.14 The figure shows the velocity graph of a passenger in an elevator.Problem 5.0 kg Correct Part C What is her weight on the moon? 89.11 Harvaran Ghai Part A An astronaut's weight on earth is 805 .1 N Correct Problem 5. . Express your answer in newtons.0 below the horizontal and the chair slides along the floor. Harvaran Ghai Part A Using Newton's laws. the magnitude of the normal force that the floor exerts on the chair. calculate .0 directed at an angle of 41. the floor is not frictionless. =143 Correct . You push on the chair with a force of = 35.Harvaran Ghai Part A What is the passenger's apparent weight at 1040 N Correct ? Part B At t = ? 735 N Correct Part C At t = ? 585 N Correct Pushing a Chair along the Floor A chair of weight 120 lies atop a horizontal floor. The board rests on a frictionless horizontal surface. Throughout the problem. and . . . as usual. In the hints.Board Pulled Out from under a Box A small box of mass is sitting on a board of mass and length . the constant force with the least magnitude that must be applied to the board in order to pull the board out from under the the box (which will then fall off of the opposite end of the board). represents the normal force exerted on . less than . The coefficient of kinetic friction between the board and the box is. for the Harvaran Ghai Part A Find . The coefficient of static friction between the board and the box is . use magnitude of the friction force between the board and the box. use for the magnitude of the acceleration due to gravity. The coefficients of static and kinetic friction between the drawbridge and the dancer's foot are and . = Correct . . Express your answer in terms of some or all of the variables answer. Do not include in your Friction Force on a Dancer on a Drawbridge A dancer is standing on one leg on a drawbridge that is about to open. respectively. and/or . The drawbridge stops just at the point where the dancer is on the verge of slipping. = Correct of the frictional force now? . The dancer is standing still on one leg. =0 Correct This shows a very important point. it is perfectly level with the ground. we can assume that the bridge is a perfectly flat surface and lacks the curvature characteristic of most bridges. When you are not told that an object is slipping or on the verge of slipping. Under these circumstances the friction force is limited by or but is otherwise not necessarily related to or . The angle should not appear . and/or . then the friction force is determined using Newton's laws of motion in conjunction with the observed motion and the other forces on the object. ? Express your answer in terms of some or all of the variables . Part B The drawbridge then starts to rise and the dancer continues to stand on one leg. Harvaran Ghai Part A Before the drawbridge starts to open. and represents the gravitational force exerted on the dancer. in your answer.the dancer by the bridge. For all the questions. What is the magnitude Express your answer in terms of some or all of the variables . . as shown in the drawing . What is the x component of the friction force. The angle should not appear = Correct Part D The bridge starts to come back down again. Harvaran Ghai Part A At the beginning of her fall.Part C Then. This half a meter corresponds to an angle degree (see the diagram. shows the sky diver (not to scale) with her position. and/or . What is the force of friction Express your answer in terms of some or all of the variables .80 for the magnitude of the acceleration due to gravity. Correct This Error! Hyperlink reference not valid. This causes the person to start to slide down the bridge at a constant speed. However. does the sky diver have an acceleration? No. and acceleration graphed as functions of time. = Correct . . Throughout this problem use 9. because the bridge is old and poorly designed. rather it stops half a meter short. Yes and her acceleration is directed upward. in your answer. Skydiving A sky diver of mass 80. You can see how her acceleration drops to zero over time. and/or now? . Yes and her acceleration is directed downward. it falls a little bit and then jerks. What is the magnitude of the frictional force now? Express your answer in terms of some or all of the variables . The dancer stops sliding. giving constant speed after a long time. again because of the age and design of the bridge it never makes it all the way down. . which has the angle exaggerated). the sky diver falls at constant speed. .0 (including parachute) jumps off a plane and begins her descent. speed. . the sky diver keeps falling at constant speed. the drag force acting on it due to air resistance can be expressed as . sky divers often will orient themselves "bellyfirst. she deploys her parachute to ensure a safe landing. =784 Correct Part C For an object falling through air at a high speed ." In this position. =56. the numerical value for Using this value for is about 0.250 . When oriented in a headfirst dive. For maximum drag and stability. (Usually the parachute is deployed when the sky diver reaches an altitude of about 900 after deploying the parachute.0 Correct Recreational sky divers can control their terminal speed to some extent by changing their body posture. where the coefficient depends on the shape and size of the falling object and on the density of air. Part D When the sky diver descends to a certain height from the ground. of the sky diver? Express you answer in meters per second. For a human body. does the skydiver have a nonzero acceleration? No. Yes and her acceleration is directed downward. the sky diver reaches her terminal speed. What is the magnitude of the drag force due to air resistance that acts on the sky diver when she has reached terminal speed? Express your answer in newtons.) Immediately . Correct Part E 3000 . what is the terminal speed .Part B At some point during her free fall. a sky diver can reach speeds of about 54 meters per second (120 miles per hour). Yes and her acceleration is directed upward. their terminal speed is typically around 45 meters per second (100 miles per hour). will be parallel to and is another vector that can lie in any direction. Alternatively. If this seems slow based on video or reallife sky divers you have seen. In picturing this vector derivative you can think of the derivative of a vector as an instantaneous quantity by thinking of the velocity of the tip of the arrow as the vector changes in time. Obviously the difference between in the same direction. When motion can occur in two dimensions (e. What is the drag force Express your answer in newtons. is confined to a tabletop but can lie anywhere in the xy plane). On the other hand.61 Correct A typical "student" parachute for recreational skydiving has a drag coefficient that gives a terminal speed for landing of about 2 meters per second (5 miles per hour). In one dimensional (straight line) motion. the effective drag coefficient of the sky diver plus parachute increases to 60.When the parachute is fully open. if has the same magnitude as . An Object Accelerating on a Ramp Learning Goal: Understand that the acceleration vector is in the direction of the change of the velocity vector. acceleration is accompanied by a change in speed. and the acceleration is always parallel (or antiparallel) to the velocity. =3.0 . you can (for small approximate the acceleration as ) . If it is longer but .88×105 Correct Part F What is the terminal speed of the sky diver when the parachute is opened? Express your answer in meters per second. acting on the sky diver immediately after she has opened the parachute? =1. that may be because the sky divers you saw were using highperformance parachutes.g. the definition of acceleration is in the limit . these offer the sky divers more maneuverability in the air but increase the terminal speed up to 4 meters per second (10 miles per hour). In general. The following questions concern the direction of the object's acceleration vector. The object is already moving along the ramp toward position 2 when it is at position 1. This problem contains several examples of this.but is in a slightly different direction.) Harvaran Ghai Part A Which direction best approximates the direction of when the object is at position 1? straight up downward to the left downward to the right straight down Correct Part B Which direction best approximates the direction of when the object is at position 2? straight up upward to the right . hence will be perpendicular to . (This is a result of the equation given above. you should find the direction of the acceleration vector by drawing the velocity vector at two points near to the position you are asked about. can have any direction relative to can differ from . its velocity vector at a point will be tangent to the track at that point. The acceleration vector will point in the same direction as the vector difference of the two velocities. . Note that since the object moves along the track. then in both magnitude and direction. In this problem.Consider an object sliding on a frictionless ramp as depicted here. where is in s.00 m/s Correct Part B What is the particle's speed at = 4. Part C Which direction best approximates the direction of when the object is at position 3? upward to the right to the right straight down downward to the right Correct Problem 6.3 A particle's trajectory is described by and . at 270 counterclockwise from the +x axis Correct 0 ? .straight down downward to the left Correct Even though the acceleration is directed straight up. Harvaran Ghai Part A What is the particle's speed at t 2.50 ? 12. this does not mean that the object is moving straight up. measured from the xaxis.6 m/s Correct Part C What is the particle's direction of motion. . . It has an initial speed . Harvaran Ghai Part A Find the time Express = Correct it takes the projectile to reach its maximum height. measured from the xaxis. at = 4. A projectile is fired from ground level at time .4 counterclockwise from the +x axis Correct Projectile Motion Tutorial Learning Goal: Understand how to apply the equations for 1dimensional motion to the y and x directions separately in order to derive standard formulae for the range and height of a projectile.50 ? 11.Part D What is the particle's direction of motion. In this problem we are assuming that the ground is level. at an angle with respect to the horizontal. and (the magnitude of the acceleration due to gravity). in terms of . a projectile that lands on a hill. The cannon is at height ball is fired with initial horizontal speed . . and the . Find the time of flight from the ymotion 3. or a projectile that must hit a moving target. Express the range in terms of = Correct . . and . Consider the x and y motion separately. and . Part C Find . Horizontal Cannon on a Cliff A cannonball is fired horizontally from the top of a cliff. above ground level.Part B Find . . and . 2. the second half of the trajectory). Express the maximum height in terms of = Correct . Find the xposition at the end of the flight this is the range. the maximum height attained by the projectile. find where the projectile lands. The actual formula for is less important than how it is obtained: 1. Express the time in terms of = Correct . If you remember these steps. such as: a cannon on a hill that fires horizontally (i. Part D Find the total distance (often called the range) traveled in the x direction. in other words.e. you can deal with many variants of the basic problem. the time at which the projectile hits the ground. Part C What is the y position of the cannonball when it is a distance Express the position of the cannonball in terms of = Correct only. Correct = . . as shown. find the initial speed of the projectile.Harvaran Ghai Part A Assume that the cannon is fired at and that the cannonball hits the ground at time position of the cannonball at the time ? Express the y position of the cannonball in terms of answer. . from the hill? . . The quantities and should not appear in your Part B Given that the projectile lands a distance Express the initial speed in terms of = Correct from the cliff. and . What is the y . Suppose that the ) with respect to the line from A to C.Problem 6. from point A to point B.612 km/h Correct Crossing a River A swimmer wants to cross a river.0 down a river.785 . what speed . as indicated in Harvaran Ghai Part A To swim directly from A to B. and the speed of the current in the river is 5 (0. The distance distance (from C to B) is 150 swimmer makes an angle of the figure.04 Correct .13 A boat takes 3. Harvaran Ghai Part A How fast is the river flowing? 0. =4. to three significant figures.70 to travel 15.30 to return. relative to the water. the . (from A to C) is 200 . should the swimmer have? Express the swimmer's speed numerically. then 5. in units of kilometers per hour. Assume that the bullet must travel through the set of disks within a single revolution. If required. use . = Correct .Another way to do this problem. without using any kinematics. as shown. and set the angle that this vector makes with AC or the river bank equal to that which AB makes with the same. apart to a shaft turning with a rotational period Harvaran Ghai Part A Derive a formula for the bullet speed in terms of . and a measured angle between the position of the hole in the first disk and that of the hole in the second. the other is down. Speed of a Bullet A bullet is shot through two cardboard disks attached a distance . would be to add the swimmer's and river's velocities vectorially. measures the angular displacement between the two holes. . for instance. means that the holes are in a line and means that when one hole is up. not its numeric equivalent. Both of the holes lie at the same radial distance from the shaft. is tangent to the bob's trajectory. is equal to the instantaneous rate of change in velocity.Direction of Acceleration of Pendulum Learning Goal: To understand that the direction of acceleration is in the direction of the change of the velocity. is perpendicular to the bob's trajectory. The eight labeled arrows represent directions to be referred to when answering the following questions. Correct Part B What is the direction of when the pendulum is at position 1? Enter the letter of the arrow parallel to . The pendulum shown makes a full swing from to . . which is unrelated to the direction of the velocity. is equal to the acceleration due to gravity. H Correct . Harvaran Ghai Part A Which of the following is a true statement about the acceleration of the pendulum bob. Ignore friction and assume that the string is massless. The road is banked at an angle . Part D What is the direction of when the pendulum reaches position 3? Give the letter of the arrow that best approximates the direction of . A cross section of the curve is shown in the diagram.Part C What is the direction of at the moment the pendulum passes position 2? Enter the letter of the arrow that best approximates the direction of . . F Correct Part E As the pendulum approaches or recedes from which position(s) is the acceleration vector almost parallel to the velocity vector . position 2 only positions 1 and 2 positions 2 and 3 positions 1 and 3 Correct Banked Frictionless Curve. C Correct We know that for the object to be traveling in a circle. and Flat Curve with Friction A car of mass traveling at speed enters a banked turn covered with ice. some component of its acceleration must be pointing radially inward. and there is no friction between the road and the car's tires. Harvaran Ghai Part A What is the radius of the turn (assuming the car continues in uniform circular motion around the turn)? Express your answer in terms of some or all of the variables acceleration due to gravity . circular road with a radius of 45 . You decide to take the race car to a small test track to experimentally determine the coefficient of friction. as well as the magnitude of the At the Test Track You want to test the grip of the tires on your new race car. = Correct . acceleration due to gravity . . the minimum value of the coefficient of static friction between the tires and the road required to prevent the car from slipping? Assume that the car's speed is still and that the radius of the curve is given by . What is . Express your answer in terms of some or all of the variables . = Correct . . as well as the magnitude of the Part B Now. The racetrack consists of a flat. suppose that the curve is level ( ) and that the ice has melted. . so that there is a coefficient of static friction between the road and the car's tires. . must the bob have so that it moves in a horizontal circle with the string always making an angle from the vertical? Express your answer in terms of some or all of the variables gravity . .. = Correct . You are to investigate the motion in which the string moves in a cone with halfangle . =0. and . Harvaran Ghai Part A What tangential speed. Error! Hyperlink reference not valid. it is a pendulum). Part A What is .. shows the result of driving the car around the track at various speeds. the coefficient of static friction between the tires and the track? Express your answer to two significant figures.95 Correct Harvaran Ghai Conical Pendulum I A bob of mass is suspended from a fixed point with a massless string of length (i. as well as the acceleration due to .e. 0 north of east.18 Part A What is the acceleration due to gravity of the sun at the distance of the earth's orbit? Harvaran Ghai 6.Part B How long does it take the bob to make one full revolution (one complete trip around the circle)? Express your answer in terms of some or all of the variables gravity . Harvaran Ghai Part A . the car's acceleration is 3.23 A car speeds up as it turns from traveling due south to heading due east.00×10−3 Correct Problem 7. When exactly halfway around the curve.0 radius of curvature.19 The passengers in a roller coaster car feel 50% heavier than their true weight as the car goes through a dip with a 40.0 Correct Problem 7. . Harvaran Ghai Part A What is the car's speed at the bottom of the dip? 14.40 . as well as the acceleration due to Correct Problem 7. 40. and . . 39 Correct Part B What is the tangential component of the acceleration at that point? 0.296 Correct A Satellite in Orbit A satellite used in a cellular telephone network has a mass of 1850 700 and is in a circular orbit at a height of above the surface of the earth.97×1024 .67×10−11 and the radius of the Earth to be = 6. .47×104 Correct Part B What fraction is this of the satellite's weight at the surface of the earth? Take the freefall acceleration at the surface of the earth to be = 9.811 Correct . .What is the radial component of the acceleration at that point? 3.38×106 . the mass of the earth to be Express your answer in newtons. = 5.80 0. =1. Harvaran Ghai Part A What is the gravitational force on the satellite? Take the gravitational constant to be = 6. the average density of the earth in kilograms per cubic meter. . .Although it is easy to find the weight of the satellite using the constant acceleration due to gravity. Notice that an astronaut's weight is never zero. and . inside the planet. it is instructional to consider the weight calculated using the law of gravitation: . Dividing the gravitational force on the satellite by . the universal gravitational constant. . . . rather than being proportional to center of the planet. for . the gravitational acceleration at the surface of the planet. When people speak of "weightlessness" in space. =5497 . an astronaut in an orbit at the same altitude.. . The distance from the planet's center to its surface (i. This assures that it is zero at the Part C Find a numerical value for . the planet's radius) is . as required by symmetry. and . Notice that increases linearly with . say.e. in terms of Express your answer in terms of = Correct .) . we find that the ratio of the forces due to the earth's gravity is simply the square of the ratio of the earth's radius to the sum of the earth's radius and the height of the orbit of the satellite above the earth. where located inside of the planet." Gravitational Acceleration inside a Planet Consider a spherical planet of uniform density . Use the radius of the earth. times a . (The object is Harvaran Ghai Part A Find an expression for the magnitude of the acceleration due to gravity. what they really mean is "free fall. An object is located a distance from the center of the planet. This will also be the fraction of the weight of. = Correct Part B Rewrite your result for function of R. Calculate your answer to four significant digits. Express the acceleration due to gravity in terms of . and a value of at the surface of . Whatever the physical cause of the interaction. the force on body A from body B is equal in magnitude and opposite in direction to the force on body B from body A. (translation by Cajori) The phrase after the colon (often omitted from textbooks) makes it clear that this is a statement about the nature of force. If you haven't learned about momentum. Newton states that the word "action" denotes both (a) the force due to an interaction and (b) the changes in momentum that it imparts to the two interacting bodies. due to gravity.Correct Newton's 3rd Law Discussed Learning Goal: To understand Newton's 3rd law. true false Correct Part B The two forces in each pair act in opposite directions. Newton wrote: To every action there is always opposed an equal reaction: or. which states that a physical interaction always generates a pair of forces on the two interacting bodies. bodies touching. you are not to assume that these two bodies have the same mass. Part A Every force has one and only one 3rd law pair force.. and directed to contrary parts. Each pairwise interaction produces a pair of opposite forces.g. true false Correct Part C Harvaran Ghai . the mutual actions of two bodies upon each other are always equal. If a statement refers to "two bodies" interacting via some force. one acting on each body. Mark each of the following statements as true or false. The central idea is that physical interactions (e. or electric forces) cause forces to arise between pairs of bodies. don't worry. each physical interaction between two bodies generates a pair of forces. In summary. for now this is just a statement about the origin of forces. Incidentally. In Principia. true false Correct Part D The two forces in each pair may have different physical origins (for instance. one of the forces could be due to gravity. smaller in magnitude and antiparallel to the force on the earth due to the moon.. and its pair force could be due to friction or electric charge). (Assume no other forces act on either body. one on each body.e. . true false Correct Part F Given that two bodies interact via some force. Part G According to Newton's 3rd law. It offers you a way to test for real forces (i. equal in magnitude and parallel to the force on the earth due to the moon.The two forces in each pair can either both act on the same body or they can act on different bodies. the accelerations of these two bodies have the same magnitude but opposite directions. equal in magnitude but antiparallel to the force on the earth due to the moon. the force on the (smaller) moon due to the (larger) earth is greater in magnitude and antiparallel to the force on the earth due to the moon. gravity) operates between two interacting bodies and generates a pair of opposite forces.. those that belong on the force side of )there should be a 3rd law pair force operating on some other body for each real force that acts on the body whose acceleration is under consideration. true false Correct Part E The two forces of a 3rd law pair always act on different bodies. greater in magnitude and parallel to the force on the earth due to the moon.) true false Correct Newton's 3rd law can be summarixed as follows: A physical interaction (e.g. inertia Correct Part B An upward force of magnitude _____ is exerted on the _____ by the table. Correct A Book on a Table A book weighing 5 N rests on top of a table.smaller in magnitude and parallel to the force on the earth due to the moon. 5 N / book Correct Part C Do the downward force in Part A and the upward force in Part B constitute a 3rd law pair? yes no . Harvaran Ghai Part A A downward force of magnitude 5 N is exerted on the book by the force of the table gravity . Its direction is _____. the net force on it must be zero (1st or 2nd law). Its direction is _____ . exerted on the _____ by the _____. There is no friction between the wedge and the block or between the wedge and the horizontal surface. Part G Which of Newton's laws dictates that the forces in Parts B and E are equal and opposite? Newton's 1st or 2nd law Newton's 3rd law Correct Block on an Incline Adjacent to a Wall A wedge with an inclination of angle rests next to a wall. . exerted on the _____ by the _____. as shown.Correct Part D The reaction to the force in Part A is a force of magnitude _____. 5 N / earth / book / upward Correct Part E The reaction to the force in Part B is a force of magnitude _____. This means that the force exerted on it by the earth must be equal and opposite to the force exerted on it by the table. A block of mass is sliding down the plane. 5 N / table / book / downward Correct Part F Which of Newton's laws dictates that the forces in Parts A and B are equal and opposite? Newton's 1st or 2nd law Newton's 3rd law Correct Since the book is at rest. Harvaran Ghai Part A Find the magnitude. along with any necessary constants. and all horizontal forces are small. This is what we should expect. of the sum of all forces acting on the block. PSS 8. the block is accelerating very slowly. Part B Find the magnitude. the block simply falls vertically and exerts no horizontal force on the wedge. we see that radians). In the second limit ( about 90 degrees). Express = Correct . of the force that the wall exerts on the wedge. Your answer to Part B could be expressed as either as gets very small or as approaches 90 degrees ( or . In either form. in terms of and . in the first limit ( small). Express = Correct .1: Dashing up the Slope . in terms of and . the contact force between the wall and the wedge goes to zero. along with any necessary constants. is reasonable. the slope B. It turns out that the girl can pull the sled up the slope with acceleration up to without slipping down the slope. VISUALIZE: Pictorial representation: Show important points in the motion with a sketch. then use kinematics to find velocites and positions. Make simplifying assumptions. ASSESS: Check that your result has the correct units. Assume that the rope connecting the girl and the sled is kept parallel to the slope at all times. Harvaran Ghai MODEL: Identify which objects are systems and which are part of the environment. Use subscript labels to distinguish forces. SOLVE: Use Newton's 2nd and 3rd laws: Write the equations of Newton's 2nd law for each system using the force information from the freebody diagrams.1 for problems involving the dynamics of an interacting systems of objects. Connect the force vectors of actionreaction pairs with dotted lines. A girl of mass is walking up a slippery slope while pulling a sled of unknown mass. Part A Which of the following objects qualify as systems in this problem? A. the girl . You may want to give each system a separate coordinate system. and answers the question. the friction between the sled and the slope is negligible. The coefficient of static friction between the girl's boots and the slope is . Solve for the acceleration. and other quantitative information relevant to the problem. such as and . the slope makes an angle with the horizontal. Include the acceleration constraints. Draw a separate freebody diagram for each system.Learning Goal: To practice ProblemSolving Strategy 8. Find the mass of the sled . the friction model. Define symbols and identify what you are trying to find. Include acceleration constraints as part of the pictorial model. Equate the magnitudes of actionreaction pairs. Physical representation: Identify all forces acting on each system and all actionreaction pairs. that act independently on more than one system. Model Start by making simplifying assumptions appropriate for the situation. The tension in the rope connecting the sled and the girl is zero. BD Correct The slope. List alphabetically all the letters corresponding to reasonable assumptions. The air resistance acting on the girl is negligible. if you think the slope and sled qualify as systems. The sled has smaller acceleration than the girl. the earth. For instance. The air resistance acting on the sled is negligible. it is not important to keep track of all the forces acting on elements of the environment. E. ACE Correct Part C Which of the following simplifying assumptions are reasonable? A. E. Do not use commas. ABD Correct . C. the air List alphabetically all the letters corresponding to the systems. C. the earth D. Unlike for the two systems. For instance. List alphabetically all the letters corresponding to reasonable assumptions. D. Do not use commas.C. however. The weight of the sled increases as the sled accelerates. F. The sled has greater acceleration than the girl. B. the sled E. Part B Which of the following simplifying assumptions are reasonable? A. They each exert external forces on the two systems (the sled and the girl). The rope connecting the sled and the girl is massless. For instance. The weight of the sled is a constant. if you think A and B are reasonable. and the air all qualify as part of the environment. type AB. type AD. The air resistance acting on the girl equals the force of friction acting on her. B. Do not use commas. D. The sled has the same acceleration as the girl. F. The normal force acting on the sled is negligible. type AB. if you think A and B are reasonable. The rope connecting the sled and the girl is unstretchable. they should be labeled on your diagram. assume that the slope angles downhill to the left: . however. You are looking for the correct number of forces in the correct directions.Visualize Now draw a sketch that includes the freebody diagrams for each system and the appropriate coordinate system. Use your sketch to answer the following questions. . Don't worry about relative magnitudes at this point. Part D Which freebody diagram for the girl is correct? Note that the forces are not labeled. they should be labeled on your diagram. a b c d e Correct Part E Which freebody diagram for the sled is correct? Note that the forces are not labeled. however. Don't worry about relative magnitudes at this point. For all questions. You are looking for the correct number of forces in the correct directions. a b c d e Correct Part F Which of these coordinate systems is most convenient for solving this problem? (The same coordinate system is appropriate for both the sled and the girl.) a b c d . The weight of the girl and the weight of the sled D.Correct Part G You should have identified the pairs of actionreaction forces on your freebody diagrams. if you think A and B are both valid actionreaction pairs. which is part of the environment. The force of friction on the girl and the tension of the string E. However. The weight of the sled and the tension of the string F. For instance. In our situation. The weight of the sled and the normal force on the sled C. F Correct An actionreaction pair between objects A and B is always a pair of forces and . Do not use commas. The weight of the sled and the gravitational force applied by the sled to the earth List alphabetically all the letters corresponding to the actionreaction pairs in this problem. type AB. then the forces "girl on the sled" and "sled on the girl" would form an actionreaction pair. the reaction force to the weight of the girl is the gravitational force applied by the girl to the earth. Each of the other forces mentioned in this question does have a reaction force. Which of the following pairs of forces form actionreaction pairs. if we assume that the girl and the sled act directly on each other (a reasonable assumption since the mass of the string is negligible). The weight of the girl and the normal force on the girl B. the objects on which such reaction forces are acting are part of the environment: For instance. of course. . according to Newton's 3rd law? A. = Correct Assess When you work on a problem on your own.There are many variations on how you might draw good pictorial and physical representations for this problem. only you can assess whether your answer seems right. Here is one example. the magnitude of the acceleration due to gravity. The following questions will help you practice the skills necessary for such an assessment. what would happen if there were very little (or no) static friction between the girl and the slope? . Part I Intuitively. without the computerprovided feedback. Express the sled's mass in terms of the given quantities and . Solve Now use the information and the insights that you have accumulated to construct the necessary mathematical expressions and to derive the solution. Part H Find the mass of the sled . Very little force would be required to pull a heavy sled up the slope. The girl would be able to pull the sled up only at constant velocity. B. Correct If the slope is horizontal. what would happen if the "slope" were horizontal and the mass of the sled were equal to the mass of the girl? The girl would be able to pull the sled with acceleration greater than . The girl would be able to pull the sled only at constant velocity. C. The girl would slip along the surface and not be able to pull the sled. Since the mass of the sled must be positive. Your formula then becomes . The girl would slip down the slope and never be able to pull the sled up. no matter how small the mass of the sled. D. and . the formula you derived in the Solve section would give a negative value for the mass of the sled. it follows that the maximum acceleration is Part K Which of the following expressions have the dimensions of mass? A. The girl would be able to pull the sled with up to some maximum acceleration that depends on the friction between her and the slope. if . The girl would be able to pull the sled up the slope with a very large acceleration. . . Correct If is very close to or equal to zero. such an answer simply means that the formula is not applicable: The girl would not be able to pull the sled up the slope. Part J Intuitively. and. List alphabetically all the letters corresponding to expressions with the correct dimensions. block has mass . Assume that each Harvaran Ghai Part A What is the magnitude of the force? Express the magnitude of the force in terms of . ACE Correct Note that trigonometric functions and the coefficient of static friction are dimensionless: They do not affect the dimension or units of the final answer.E. Pulling Three Blocks Three identical blocks connected by ideal strings are being pulled along a horizontal frictionless surface by a horizontal force . type AB. Do not use commas. The magnitude of the tension in the string between blocks B and C is . = Correct . For instance. if you think A and B have the correct dimensions. . Once block B is set into downward motion. also of weight . Assume that the mass and friction of the pulley are negligible. as shown. = Correct Part B A cat. what is the magnitude of its acceleration? = Correct Two Masses. If block B is now set into downward motion. and an Inclined Plane Block 1. of mass . it descends at a constant speed. falls asleep on top of block A. Block A has weight and block B has weight . of mass .Kinetic Friction in a Block-and-Pulley System Consider the system shown in the figure . a Pulley. is connected over an ideal (massless and frictionless) pulley to block 2. Harvaran Ghai Part A Calculate the coefficient of kinetic friction between block A and the table top. Assume that the blocks accelerate as shown with an acceleration of magnitude and that the coefficient of kinetic friction between block 2 and the plane is . then integration of both sides of equation (1) gives . and . and is the rate at which the object's momentum is changing. Express your answer in terms of some or all of the variables . (2) .Harvaran Ghai Part A Find the ratio of the masses . . = Correct The Impulse-Momentum Theorem Learning Goal: To learn about the impulsemomentum theorem and its applications in some common cases. as well as the magnitude of the acceleration due to gravity . (1) where is the net force acting on the object. If the object is observed during an interval of time between times and . Using the concept of momentum. Newton's second law can be rewritten as . the impulsemomentum theorem can be written as acting along the . Express your answer in terms of any or all of = Correct Part B . it is often useful to apply equation (3) to the x and y components of motion separately. Harvaran Ghai The following questions will help you learn to apply the impulsemomentum theorem to the cases of constant and varying force acting along the direction of motion. In the case of a constant net force direction of motion. . is a constant force. and are the components of the corresponding vector quantities along the chosen coordinate axis.The right side of equation (2) is simply the change in the object's momentum . (3) Here . If the motion in question is twodimensional. The net force is acting on the particle along the x axis. . It states that the change in an object's momentum is equal to the impulse of the net force acting on the object. let us consider a particle of mass moving along the x axis. Part A The particle starts from rest at that . and . . The left side is called the impulse of the net force and is denoted by . This equation is known as the impulsemomentum theorem. Then equation (2) can be rewritten as . First. What is the magnitude of the momentum of the particle at time ? Assume . Express your answer in terms of any or all of = Correct . the magnitude of its velocity . Let us now consider several twodimensional situations. A particle of mass is moving in the positive x direction at speed . After a certain constant force is applied to the particle. . What is the magnitude of the velocity of the particle at time ? Assume that . = Correct Part F Which of the vectors below best represents the direction of the impulse vector ? .The particle starts from rest at . it moves in the positive y direction at speed . Part D The particle has momentum of magnitude seconds later? at a certain instant. Part E Find the magnitude of the impulse delivered to the particle. the magnitude of its momentum . . What is Express your answer in terms of any or all of = Correct . and . Part C The particle has momentum of magnitude seconds later? at a certain instant. and . What is Express your answer in terms of any or all of = Correct . . . Express your answer in terms of and . . Use three significant figures in the numerical coefficient. and . So far. in the positive y direction? Express your answer in terms of = Correct . .6 degrees 30 degrees 60 degrees 63. If you use a numerical coefficient. A varying force is acting on the particle between speed of the particle at seconds. Part I A particle of mass kilograms is at rest at seconds.4 degrees Correct Part H If the magnitude of the net force acting on the particle is . how long does it take the particle to acquire its final velocity. and . we have considered only the situation in which the magnitude of the net force acting on the particle was either irrelevant to the solution or was considered constant. Find the . Let us now consider an example of a varying force acting on a particle. use three significant figures.1 2 3 4 5 6 7 8 Correct Part G What is the angle between the positive y axis and the vector as shown in the figure? 26. seconds and seconds. 78 Correct . 2. with its engine in neutral. through the water at speed 3.167 s Correct Filling the Boat A boat of mass 250 is coasting. The rain is falling vertically. and it accumulates in the boat at the rate of 10. =43 Correct Problem 9.00 rain. Harvaran Ghai Part A What is the speed of the boat after time 2.11 A 500 airtrack glider collides with a spring at one end of the track.00 has passed? Assume that the water resistance is negligible.0 when it starts to .Express your answer in meters per second to three significant figures. Express your answer in meters per second. shows the glider's velocity Harvaran Ghai Part A How long is the glider in contact with the spring? 0. The figure and the force exerted on the glider by the spring. Is the component of the total momentum of the system parallel to the direction of motion still conserved? yes no Correct The boat is subject to an external force. The astronaut finds herself at rest relative to the spaceship.80×10−2 Correct PSS 9. An astronaut performs maintenance work outside her spaceship when the tether connecting her to the spaceship breaks. the drag force due to water resistance.1 for problems involving conservation of momentum. the mass of the wrench is . in the form . choose a system that is isolated ( ) or within which the interactions are sufficiently short and intense that you can ignore external forces for the duration of the interaction (the impulse approximation). To get back to the ship. she decides to sacrifice her favorite wrench and hurls it directly away from the spaceship at a speed relative to the spaceship. Momentum is conserved. . Part C The drag is proportional to the square of the speed of the boat. Express your answer in meters per second per second. What is the acceleration of the boat just after the rain starts? Take the positive axis along the direction of motion. What is the distance between the spaceship and the wrench by the time the astronaut reaches the spaceship? The mass of the astronaut is . Harvaran Ghai MODEL: Clearly define the system.1: Tools of the Trade Learning Goal: To practice ProblemSolving Strategy 9. If possible. −1. at a distance from it. and therefore its momentum is not conserved.Part B Now assume that the boat is subject to a drag force due to water resistance. For example. The force that the spaceship exerts on the wrench is very large. In this case. Part A In addition to the astronaut. the earth Enter the letter(s) of the correct answer(s) in alphabetical order. VISUALIZE: The mathematical representation is based on the law of conservation of momentum: form. this is SOLVE: . and identify what you are trying to find. is reasonable. Define symbols that will be used in the problem. The force that the astronaut exerts on the wrench is very large. and answers the question. The force that the astronaut exerts on the wrench is very small. Model We start by choosing the objects that would make up the system. B Correct Part B Which of the following reasons best explains why the astronaut + wrench can be considered an isolated system? The mass of the wrench is much smaller than that of the astronaut. Draw a beforeandafter pictorial representation. In component . enter ABC. Other segments of the motion can be analyzed using Newton's laws or. as you'll learn in Chapters 10 and 11. Do not use commas. list known values. ASSESS: Check if your result has the correct units. try to divide the problem into parts such that momentum is conserved during one segment of the motion. if you think the system consists of all the objects listed. Correct . the spaceship B. the wrench C. If it is not possible to choose an isolated system. which of the following are components of the system that should be defined to solve the problem? A. conservation of energy. it is possible to identify the system that is isolated. The force that the spaceship exerts on the wrench is very small. Correct Part D Which statement about . both known and unknown. Part E Find the final distance between the spaceship and the wrench. the wrench will have covered a certain distance. You may or may not use all of them. By the time the astronaut reaches the spaceship. in perpendicular directions. Solve Now use the information and the insights that you have accumulated to construct the necessary mathematical expressions and to derive the solution. label this distance . in the same direction.Visualize Now draw a beforeandafter pictorial representation including all the elements listed in the problemsolving strategy. and is correct? Correct Here is an example of what a good beforeandafter pictorial representation might look like for this problem. Be sure that your sketch is clear and includes all necessary symbols. the astronaut and the wrench move in opposite directions. . Part C After the wrench is thrown. Express the distance in terms of the given variables. . on your pictorial representation. D. if you think that only expressions C and D have the units of distance. For instance. Part G Which of the following mathematical expressions have the units of distance. If the astronaut were more massive. E. BC Correct could only be zero if . If the astronaut threw a space pen instead of a wrench. For realistic values of the quantities involved. B. (Assume the space pen weighs less than the wrench). Do not use commas. it is possible that . the wrench would travel further in the time it takes the astronaut to reach the ship. The following questions will help you practice the skills necessary for such an assessment. C. As you can see from your answer. C. Part F Intuitively. this would only happen if the mass of the astronaut were zero. type CD. which of the following statements are correct? A. which is obviously unrealistic. Type the letters corresponding to the correct answers. B. only you can assess whether your answer seems right. without the computerprovided feedback. and are distances? . where A. = Correct Assess When you work on a problem on your own. F. the pen would travel further than the wrench would in the time it takes the astronaut to reach the ship. the speed of the twocar unit after the collision. and car 2 was traveling northward at a speed of . Before the collision. Harvaran Ghai Part A First. Two cars of masses and collide at an intersection. that is. find the magnitude of . For instance. Do not use commas. ABE Correct Colliding Cars In this problem we will consider the collision of two cars initially moving at right angles. Express in terms of . The collision is therefore completely inelastic. . = Correct Part B Find the tangent of the angle . and the cars' initial speeds and . if you think that only expressions C and D have the units of distance. type CD. After the collision.Type the letters corresponding to the correct answers. . car 1 was traveling eastward at a speed of . We assume that after the collision the cars stick together and travel off as a single unit. the two cars stick together and travel off in the direction shown. both of mass . Correct Collision at an Angle Two cars. . This means that before the collision: The velocities of the cars were equal. in other words. the twocar system travels at speed at an angle east of north. . is The magnitudes of the momenta of the cars were equal. while the second was traveling at speed at an angle south of east (as indicated in the figure). Prior to the collision. After the collision. Harvaran Ghai Part A Find the speed of the joined cars after the collision. The masses of the cars were equal. Express your answer in terms of and . = Correct and . = . one car had been traveling north at speed .Express your answer in terms of the momenta of the two cars. Part C Suppose that after the collision. collide and stick together. At height For this problem. the girl grabs a box of mass kilograms.Correct Part B What is the angle with respect to north made by the velocity vector of the two cars after the collision? Express your answer in terms of . =4. Part A What is the speed of the girl immediately before she grabs the box? Express your answer numerically in meters per second. Your answer should contain an inverse trigonometric function. use kilograms springs from a trampoline with an initial upward velocity of meters above the trampoline. meters per second per second for the magnitude of the acceleration due to gravity. = Correct A Girl on a Trampoline A girl of mass second.98 Correct meters per Harvaran Ghai . with respect to the top of the =2. =3. It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion. and label the corresponding tensions in the string and . . some of the system's kinetic energy is lost. and have magnitudes and .81 Correct Circling Ball A ball of mass is attached to a string of length . At the top and bottom of the vertical circle.Part B What is the speed of the girl immediately after she grabs the box? Express your answer numerically in meters per second. Express your answer numerically in meters. Part D What is the maximum height that the girl (with box) reaches? Measure trampoline. In this case the kinetic energy lost is converted to heat energy in the girl's muscles as she grabs the box. To avoid confusion.98 Correct Part C Is this "collision" elastic or inelastic? elastic inelastic Correct In inelastic collisions. label the ball's speeds and . and sound energy. Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy. take the upward direction to be positive throughout the problem. Assume the following: The bungee cord behaves as an ideal spring once it begins to stretch. a bungee jumper. = Correct The method outlined in the hints is really the only practical way to do this problem.Harvaran Ghai Part A Find . Bungee Jumping Kate. wants to jump off the edge of a bridge that spans a river below. which has length when unstretched. . Kate doesn't actually jump but simply steps off the edge of the bridge and falls straight downward. Kate has a mass . Kate's height is negligible compared to the length of the bungee cord. Express the difference in tension in terms of and . and the surface of the bridge is a height above the water. . she can be treated as a point particle. Hence. If done properly. can be accomplished fairly simply and elegantly. will first straighten and then stretch as Kate falls. with spring constant . The bungee cord. The quantities and should not appear in your final answer. finding the difference between the tensions. the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle. . each of mass . before the first bounce). Dancing Balls Four balls. = Correct . Express the distance in terms of quantities given in the problem introduction.Use for the magnitude of the acceleration due to gravity. what is the spring constant ? Ignore all dissipative forces. are connected by four identical relaxed springs with spring constant . Express in terms of .e. = Correct Part B If Kate just touches the surface of the river on her first downward trip (i. and . The balls are simultaneously given equal initial speeds directed away from the center of symmetry of the system. Harvaran Ghai Part A How far below the bridge will Kate eventually be hanging. . Harvaran Ghai . once she stops oscillating and comes finally to rest? Assume that she doesn't touch the water. . their kinetic energy reaches __________. Assume that . of mass . . so that after the collision. while block 2 moves with speed . (It may help to draw the initial and the final states of the system. and = Correct . moves across a frictionless surface with speed . of mass . Elastic Collision in One Dimension Block 1. a maximum zero neither a maximum nor zero Correct Part B Use geometry to find . = Correct Part C Find the maximum displacement of any one of the balls from its initial position.) Express your answer in terms of . the distance each of the springs has stretched from its equilibrium position. Express in terms of some or all of the given quantities . After the collision. the two objects move off in the direction of the first object before the collision. block 1 moves with speed . the maximum displacement of each ball from its initial position. It collides elastically with block 2.Part A As the balls reach their maximum displacement. which is at rest ( ). Part A This collision is elastic. = Correct Part C What is the final speed of block 2? Express in terms of . . and . = Correct Sinking the 9-Ball Harvaran Ghai . . if any. are conserved in this collision? kinetic energy only momentum only kinetic energy and momentum Correct Part B What is the final speed of block 1? Express in terms of . and . What quantities. one of which is initially at rest. =45. as shown in the figure. so she needs to line up her shot precisely. across the room. Let When released. the car slides be the initial position of the car. She will win if she sinks the 9ball from the final rack. Furthermore. Energy in a Spring Graphing Question A toy car is held at rest against a compressed spring. Jeanette carefully hits the cue ball into the 9ball off center. . neglect friction. the angle between the final velocities of the two objects will be ninety degrees. and ignore the spinning of the balls. as shown in the figure.Jeanette is playing in a 9ball pool tournament. after the collision. while the 9ball moves at speed . It turns out that in any elastic collision between two objects of equal mass. so that when the balls collide. they move away from each other at the same angle from the direction in which the cue ball was originally traveling (see figure).0 Correct Note that the angle between the final velocities of the two balls is . the cue ball moves away at speed . and the cue ball is hit at an initial speed of . For the purposes of this problem. Assume that friction is negligible. assume that the collision is perfectly elastic. Both the cue ball and the 9ball have mass . Express your answer in degrees to three significant figures. Harvaran Ghai Part A Find the angle that the 9ball travels away from the horizontal. so your graph should simply reflect the qualitative shape of the energy vs.Harvaran Ghai Part A Sketch a graph of the total energy of the spring and car system. time plot. There is no scale given. HORIZONTAL LINE Correct . 5 POINTS GRADUALLY SLOPING UP OPPOSITE OF GIVEN . 4 POINTS GRADUALLY SLOPING DOWN Correct Part C Sketch a graph of the car's kinetic energy from the moment it is released until it passes the equilibrium position of the spring.Part B Sketch a plot of the elastic potential energy of the spring from the point at which the car is released to the equilibrium position of the spring. Make your graph consistent with the given plot of total energy (the gray line given in the graphing window). Your graph should be consistent with the given plots of total energy (gray line in graphing window) and potential energy (gray parabola in graphing window). For this problem.0 .Correct Graphing Gravitational Potential Energy A 1.00 ball is thrown directly upward with an initial speed of 16. A graph of the ball's gravitational potential energy vs. . Harvaran Ghai Part A Draw a line on the graph representing the total energy of the ball. for an arbitrary initial velocity is given in Part A. take as the acceleration due to gravity. height. The zero point of gravitational potential energy is located at the height at which the ball leaves the thrower's hand. HORIZONTAL LINE AT 126 . 5) . The graph for a 1. This occurs at the height at which the total energy and potential energy graphs intersect.8 Correct The ball reaches its maximum height when its velocity (and therefore kinetic energy) is zero. (100. Take . height graph to represent the gravitational potential energy if the ball had a mass of 2.00 reference.00 ball with an arbitrary initial velocity is provided again as a as the acceleration due to gravity. Express your answer to one decimal place. determine the maximum height reached by the ball.5) (150. Part C Draw a new gravitational potential energy vs. so all of its energy is potential.Correct Part B Using the graph. 12. 7. Harvaran Ghai . It seems reasonable to say that the speed of an objectand. we will explore the mathematical relationship between the work done on an object and the change in the kinetic energy of that object. height to have twice the slope. you should expect the plot of potential energy vs. In this problem. therefore. its kinetic energycan be changed by performing work on the object. Understanding Work and Kinetic Energy Learning Goal: To learn about the WorkEnergy Theorem and its basic applications. In this problem. you will learn about the relationship between the work done on an object and the kinetic energy of that object. The kinetic energy of an object of mass moving at a speed is defined as .Correct For a ball with twice the mass. zero negative positive . zero negative positive Correct Part C The work done on the sled by the normal force is __________. zero negative positive Correct Part E The work done on the sled by the force of friction is __________. horizontal force of magnitude along a rough. zero negative positive Correct Part D The work done on the sled by the pulling force is __________. horizontal surface. let us consider a sled of mass being pulled by a constant.First. Part A How many forces are acting on the sled? one two three four Correct Part B The work done on the sled by the force of gravity is __________. The sled is speeding up. Consider an interval of time during which the sled covers a distance and the speed of the sled increases from to . zero negative positive Correct Part G In the situation described. . . .Correct Part F The net work done on the sled is __________. Let the mass of the sled be acting on the sled be and the magnitude of the net force . Part H Find the net force acting on the sled. The sled starts from rest. We will use this information to find the relationship between the work done by the net force (otherwise known as the net work) and the change in the kinetic energy of the sled. remains constant decreases increases Correct Let us now consider the situation quantitatively. and Part I Find the net work done on the sled. Express your answer in terms of some or all of the variables = Correct and . . Express your answer in terms of some or all of the variables = Correct . the kinetic energy of the sled __________. and . Your answer can also be rewritten as or . including those involving nonconstant forces or forces acting at an angle to the displacement of the object. This theorem is quite useful in solving problems. one can write . however. the initial and the final kinetic energies of the sled. Find the total work done on the car by the external forces.Part J Use to find the net work done on the sled. This formula is known as the WorkEnergy Theorem. it can be shown that the WorkEnergy Theorem is applicable in all situations. You may or may not use all of them. Express your answer in terms of some or all of the variables = Correct . Express your answer in terms of the given quantities. respectively. as illustrated by the following example. they should not be interpreted as a proof of this theorem. = Correct . and the acceleration due to gravity is . Nevertheless. The calculations done in this problem illustrate the applicability of this theorem in a particlar case. The coefficient of static friction is . Finally. . Part K A car of mass accelerates from speed to speed while going up a slope that makes an angle with the horizontal. where and are. Here is a simple application of the WorkEnergy Theorem. one end of which is attached to a wall. = Correct Work from a Constant Force Learning Goal: To understand how to compute the work done by a constant force acting on a particle that moves in a straight line. A force is considered constant if independent of . Harvaran Ghai Part A The spring is now compressed so that the unconstrained end moves from to . is . Express the work done by the spring in terms of and . unstretched.Work Done by a Spring Consider a spring. In this problem. find the work done by the spring as it is compressed. you will calculate the work done by a constant force. This is the most frequently encountered situation in elementary Newtonian mechanics. Using the work integral . with the unconstrained end of the spring at position The spring is initially . with spring constant . Harvaran Ghai Part A Consider a particle moving in a straight line from initial point B to final point A, acted upon by a constant force . The force (think of it as a field, having a magnitude and direction at every position ) is indicated by a series of identical vectors pointing to the left, parallel to the horizontal axis. The vectors are all identical only because the force is constant along the path. The magnitude of the force is , and the displacement vector from point B to point A is (of magnitude , making and angle (radians) with the positive x axis). Find , the work that the force performs on the particle as it moves from point B to point A. Express the work in terms of , , and . Remember to use radians, not degrees, for any angles that appear in your answer. = Correct This result is worth remembering! The work done by a constant force of magnitude , which acts at an angle of with respect to the direction of motion along a straight path of length , is . This equation correctly gives the sign in this problem. Since is the angle with respect to the positive x axis (in radians), ; hence . Part B Now consider the same force acting on a particle that travels from point A to point B. vector now points in the opposite direction as it did in Part A. Find the work Express your answer in terms of , , and . The displacement done by in this case. = Correct The Work Done in Pulling a Supertanker Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.60×106 , one at an angle 14.0 west of north, and the other at an angle 14.0 east of north, as they pull the tanker a distance 0.890 north. toward the Harvaran Ghai Part A What is the total work done by the two tugboats on the supertanker? Express your answer in joules. 2.76×109 Correct Work on a Sliding Block A block of weight sits on a frictionless inclined plane, which makes an angle with respect to the horizontal, as shown. Part A A force of magnitude , applied parallel to the incline, pulls the block up the plane at constant speed. Harvaran Ghai The block moves a distance up the incline. The block does not stop after moving this distance but continues to move with constant speed. What is the total work done on the block by all forces? (Include only the work done after the block has started moving, not the work needed to start the block moving from rest.) Express your answer in terms of given quantities. =0 Correct Part B What is , the work done on the block by the force of gravity as the block moves a distance up the incline? Express the work done by gravity in terms of the weight and any other quantities given in the problem introduction. = Correct Part C What is , the work done on the block by the applied force as the block moves a distance up the incline? Express your answer in terms of and other given quantities. = Correct Part D What is , the work done on the block by the normal force as the block moves a distance up the inclined plane? Express your answer in terms of given quantities. =0 Correct Potential Energy Graphs and Motion Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a particle. Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle's kinetic energy, etc. If you are still having trouble visualizing this. Therefore. The key idea in interpreting the graph can be expressed in the equation where is the x component of the net force as function of the particle's coordinate . and its potential energy is increasing. we will use the term force instead of the cumbersome x component of the net force. The total energy is the sum of kinetic ( ) and potential ( ) energies of the particle. In answering the following questions.Consider the potential energy diagram shown. moving upward). the particle is said to be in equilibrium. the net force would be pulling the particle to the left. Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill). we will assume that there is a single varying force acting on the particle along the x axis. The curve represents the value of potential energy as a function of the particle's coordinate . if the particle is moving to the right. If the x component of the net force is zero. downward). . Note the negative sign: It means that the x component of the net force is negative when the derivative is positive and vice versa. consider the following: If a massive particle is increasing its gravitational potential energy (that is. The horizontal line above the curve represents the constant value of the total energy of the particle . There are two kinds of equilibrium: Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills). the force of gravity is pulling in the opposite direction (that is. For instance. pull the particle back to the left. directed to the right directed to the left equal to zero Correct The slope of the graph is zero. therefore. it would be "climbing the hill. more abstract way of thinking about this is to say that the slope of the graph at point A is positive. the derivative Part D The acceleration of the particle at point B is __________.Harvaran Ghai Part A The force acting on the particle at point A is __________. Part B The force acting on the particle at point C is __________. directed to the right directed to the left equal to zero Correct Consider the graph in the region of point A." that is. . and . If the particle is moving to the right. directed to the right directed to the left equal to zero Correct Part C The force acting on the particle at point B is __________." and the force would "pull it down. the direction of is negative. therefore. directed to the right directed to the left equal to zero Correct . Another. for instance.If the net force is zero. List your choices alphabetically. and E. its acceleration is __________. for instance. Part G Name all labeled points on the graph corresponding to unstable equilibrium. and E. type your answer as BDE. The particle is said to be in a state of equilibrium. D. if you choose points B. if you choose points B. List your choices alphabetically. with no commas or spaces. so is the acceleration. its acceleration is __________. small deviations from equilibrium at point B cause a force that accelerates the particle further away. BDFH Correct . BF Correct Part H Name all labeled points on the graph corresponding to stable equilibrium. and E. Part E If the particle is located slightly to the left of point B. type your answer as BDE. directed to the right directed to the left equal to zero Correct Part F If the particle is located slightly to the right of point B. directed to the right directed to the left equal to zero Correct As you can see. D. type your answer as BDE. with no commas or spaces. D. List your choices alphabetically. if you choose points B. with no commas or spaces. for instance. DH Correct Part I Name all labeled points on the graph where the acceleration of the particle is zero. hence the particle is in unstable equilibrium. AE Correct Part K Consider points A. Part J Name all labeled points such that when a particle is released from rest there. Of these three points. includes the locations of both stable and unstable equilibrium.Your answer. and G. type your answer as BDE. of course. for instance. on this graph. and E. For example. List your choices alphabetically. E. Part M At what point on the graph does the particle have the lowest speed? B Correct . D. Part L What point on the graph corresponds to the maximum kinetic energy of the moving particle? D Correct It makes sense that the kinetic energy of the particle is maximum at one of the (force) equilibrium points. which one corresponds to the greatest magnitude of acceleration of the particle? A E G Correct Kinetic energy If the total energy of the particle is known. if you choose points B. it would accelerate to the left. the total energy is shown by the horizontal line. one can also use the graph of kinetic energy of the particle since to draw conclusions about the . with no commas or spaces. think of a pendulum (which has only one force equilibrium pointat the very bottom). As a reminder. and is the unit vector in the radial direction in case of a spherically symmetrical force field. in the equations for the forces. think of a gravitational field (the one that makes apples fall down and keeps the planets orbiting) or an electrostatic field existing around any electrically charged object. If a particle is moving in such a field. It is helpful to understand the character of motion qualitatively before you attempt quantitative problems. Evaluating such an integral in a general case can be a tedious and lengthy task. is the unit vector in the x direction. many different conclusions can be made about the particle's motion merely by looking at the graph. its change in potential energy does not depend on the particle's path and is determined only by the particle's initial and final positions. This problem should prove useful in improving such an understanding. the component of the net force acting on a particle equals the negative derivative of the potential energy function along the corresponding axis: . 2. you will practice calculating the change in potential energy for a particle moving in three common force fields. Imagine that a conservative force field is defined in a certain region of space. However. The most common realworld fields are rather simply defined. where is the change in potential energy for a particle moving from point 1 to point 2.As you can see. and is a small displacement of the particle along its path from 1 to 2. Recall that. Does this sound too abstract? Well. Because the result is pathindependent. Potential Energy Calculations Learning Goal: To understand the relationship between the force and the potential energy changes associated with that force and to be able to calculate the changes in potential energy as definite integrals. the change in potential energy can be found as the integral . is the unit vector in the y direction. In this problem. Therefore. it is always possible to consider the most straightforward way to reach point 2 from point 1. two circumstances make it easier: 1. Note that. Harvaran Ghai . in general. is the net force acting on the particle at a given point of its path. and . and . The magnitude and direction of such a force are given by Newton's law of gravity: . . . Find . Part C Finally. consider the gravitational force generated by a spherically symmetrical massive object. and are constants. . . . where . where and Express your answer in terms of Correct = . Find . and .Part A Consider a uniform gravitational field (a fair approximation near the surface of a planet). Express your answer in terms of . . Correct = . Find . and the spring constant is positive. Part B Consider the force exerted by a spring that obeys Hooke's law. where . this method. . the spring constant. Dragging a Board A uniform board of length and mass lies near a boundary that separates two regions. A Mass-Spring System with Recoil and Friction An object of mass is traveling on a horizontal surface. and the potential energy is the function of the particle's position. and . the coefficient of kinetic friction between the board and the surface is . The positive direction is shown in the figure. The object compresses the spring. the change in potential energy of the particle can be found by integrating the force along the particle's path. In case of a nonconservative forcesuch as a frictional or magnetic forcethe potential energy can no longer be defined as a function of the particle's position. .Express your answer in terms of . and the method that you used in this problem would not be applicable. and in region 2. Correct = . as we mentioned before. and then recoils and travels in the opposite direction. In region 1. does have an important restriction: It can only be applied to a conservative force field. As you can see. For conservative forces such as gravity or tension the work done on the particle does not depend on the particle's path. and . When the object reaches trip. . it stops. There is a coefficient of kinetic friction between the object and the surface. The object has speed when it reaches and encounters a spring. on its return Harvaran Ghai Part A Find . However. Express in terms of . stops. . = Correct . the coefficient is . ) Express your answer in terms of = Correct . . and . it in fact did. and half in region 2. This answer makes sense because it is as if the board spent half its time in region 1. Assume that the magnitude of the acceleration due to gravity is . . . Express the net work in terms of = Correct . . Assume that the board moves at constant velocity. Part A Harvaran Ghai . . assume that the board moves at constant velocity.Harvaran Ghai Part A Find the net work done by friction in pulling the board directly from region 1 to region 2. Part B What is the total work done by the external force in pulling the board from region 1 to region 2? (Again. Drag on a Skydiver A skydiver of mass jumps from a hot air balloon and falls a distance before reaching a terminal velocity of magnitude . and . which on average. . where is the crosssectional area of the vehicle and is called the coefficient of drag.) at operating speeds is called form drag. . Find the power dissipated by form drag. cars. is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified. planes. by the drag force of the air? Express the work in terms of . Part B A certain car has an engine that provides a maximum power . Harvaran Ghai Part A Consider a vehicle moving with constant velocity . Part B Find the power supplied by the drag force after the skydiver has reached terminal velocity . = Correct Power Dissipation Puts a Drag on Racing The dominant form of drag experienced by vehicles (bikes. and the magnitude of the acceleration due to gravity . . It increases quadratically with velocity (essentially because the amount of air you run into increases with and so does the amount of force you must exert on each small volume of air). over the distance .What is the work done on the skydiver. so that the new power Assume the following: is 10 percent greater than the original power ( . = Correct . . etc. and speed . Express the power in terms of quantities given in the problem introduction. . Suppose that the maximum speed of the car. Thus . Express your answer in terms of = Correct . By what percentage. the spacecraft would eventually crash into the earth. The magnitude of the force of air drag at these speeds is proportional to the square of the speed. The top speed is limited by air drag. . a spacecraft has run out of fuel and its kinetic energy is zero. Neglect air resistance throughout this problem. . If only the gravitational force of the earth were to act on it (i. neglect the forces from the sun and other solar system objects). when the quantities are related by proportionalities of exponents. Express the speed in terms of = Correct . where . Correct =3. The mass of the earth is and its radius is . Energy of a Spacecraft Very far from earth (at ). . This dependence of small changes on each other. Part B Now find the spacecraft's speed when its distance from the center of the earth is Express the speed in terms of and . = Correct Orbiting Satellite . and the universal gravitational constant . since the spacecraft is primarily moving through the near vacuum of space. is common in physics and often makes a useful shortcut for estimations.. is the top speed of the car increased? Express the percent increase in top speed numerically to two significant figures.e.2 % You'll note that your answer is very close to onethird of the percentage by which the power was increased. Harvaran Ghai Part A Find the speed of the spacecraft when it crashes into the earth. .) Use made of a material with for the universal gravitational Harvaran Ghai Part A Find the kinetic energy of this satellite. not its surface. Express the satellite's kinetic energy in terms of . . Part C What is the ratio of the kinetic energy of this satellite to its potential energy? Express in terms of parameters given in the introduction. Express the satellite's gravitational potential energy in terms of = Correct . . . = Correct . . . . . . is in a circular orbit of radius around a spherical planet of radius is measured from the center of the planet.A satellite of mass density . Take the gravitational potential energy to be zero for an object infinitely far away from the planet. and . the gravitational potential energy of the satellite. ( constant. Part B Find . and . or masses tied together with springs (where since the force law is ). Thus it applies to stars in a galaxy. It needs to reach a 700 Hubble Space Telescope for repairs. Harvaran Ghai Part A How much energy is required to boost it to the new orbit? 1. The shuttle's mass is 8. Harvaran Ghai Part A . This is a specical case of a general and powerful theroem of advanced classical mechanics known as the Virial Theorem.00×104 high circular orbit to catch the . so that the angular velocity of his tires follows the equation . ).53×1011 J Correct An Exhausted Bicyclist An exhausted bicyclist pedals somewhat erratically. The theorem applies to the average of the kinetic and potential energies of of any one or multiple objects moving over any closed (or almost closed) path that returns very close to itself provided that all objects interact via potentials with the same power law dependence on their separation. where represents time (measured in seconds). Problem 12.28 The space shuttle is in a 250 high circular orbit.500 Correct The result of this problem may be expressed as where is the exponent of the force law (i. =0.e. =45. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is decreasing. Take the position of the spot at time to be at angle radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and measure positive angles in the direction of the tire's rotation. The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is increasing.793 Correct Part B Express the angular displacement undergone by the spot of paint at seconds in degrees. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is increasing.There is a spot of paint on the front tire of the bicycle.5 Correct Part C What distance has the spot of paint moved in 2 seconds if the radius of the tire is 50 centimeters? Express your answer in centimeters. Correct A Spinning Grinding Wheel . =39. What angular displacement has the spot of paint undergone between time 0 and 2 seconds? Express your answer in radians. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is increasing. =0.7 Correct Part D Which one of the following statements describes the motion of the spot of paint at seconds? The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is decreasing. 0 32. 8. where is the angle between and . From then on. Harvaran Ghai Part A Through what total angle did the wheel turn between Express your answer in radians.At time a grinding wheel has an angular velocity of 23. 12. For this problem . and D) all lie in the xy plane.11 Correct Finding Torque A force of magnitude . . B.2 Correct Part C What was the wheel's angular acceleration as it slowed down? Express your answer in radians per second per second. The torque of a force acting on a particle having a position vector with respect to a reference point (thus points from the reference point to the point at which the force acts) is equal to the cross product of and . The magnitude of the torque is . It has a constant angular acceleration of until a circuit breaker trips at time = 1.. negative torque about a reference point corresponds . making an angle with the x axis. The vector and the four reference points (i.e. 0) in the figure. the wheel turns through an angle of as it coasts to a stop at constant angular deceleration. C.90 .0 433 . A. Rotation axes A D lie parallel to the z axis and pass through each respective reference point. the direction of is perpendicular to both and . and the time it stopped? 534 Correct Part B At what time does the wheel stop? Express your answer in seconds. at Cartesian coordinates (0. is applied to a particle located at point A. You must express in terms of . .to clockwise rotation. . Harvaran Ghai Part A What is the torque due to force about the point A? Express the torque about point A at Cartesian coordinates (0. and/or other given coordinate data. . 0). . a distance Express the torque about point C in terms of . . . located at a position given by Cartesian coordinates ( . = Correct .) Express the torque about point B in terms of . and/or when entering your answers. . = Correct Part C What is the torque along the x axis? about the point C. =0 Correct Part B What is the torque due to force about the point B? (B is the point at Cartesian coordinates (0. ). and/or other given coordinate data. 0). located a distance from the origin along the y axis. and a small sphere of mass is attached to the right end. . The gravitational force acts downward. Part A Harvaran Ghai . and/or other given coordinate data. with the magnitude of the gravitational acceleration equal to . as shown in the diagram. A small sphere of mass is attached to the left end of the rod. = Correct Note that the cross product which simplifies to can also be expressed as a thirdorder determinant when and lie in the xy plane.Part D What is the torque axis? about the point D. . located at a distance from the origin and making an angle with the x Express the torque about point D in terms of . . The spheres are small enough that they can be considered point particles. Pivoted Rod with Unequal Masses A thin rod of mass and length is allowed to pivot freely about its center. What is the moment of inertia of this assembly about the axis through which it is pivoted? Express the moment of inertia in terms of = Correct , , , and . Remember, the length of the rod is , not . Part B Suppose the rod is held at rest horizontally and then released. (Throughout the remainder of this problem, your answer may include the symbol , the moment of inertia of the assembly, whether or not you have answered the first part correctly.) What is the angular acceleration of the rod immediately after it is released? Take the counterclockwise direction to be positive. Express in terms of some or all of the variables , , , , , and . = Correct Pulling a String to Accelerate a Wheel A bicycle wheel is mounted on a fixed, frictionless axle, as shown . A massless string is wound around the wheel's rim, and a constant horizontal force of magnitude starts pulling the string from the top of the wheel starting at time when the wheel is not rotating. Suppose that at some later time the string has been pulled through a distance . The wheel has moment of inertia , where is a dimensionless number less than 1, is the wheel's mass, and is its radius. Assume that the string does not slip on the wheel. Harvaran Ghai Part A Find , the angular acceleration of the wheel, which results from pulling the string to the left. Use the standard convention that counterclockwise angular accelerations are positive. Express the angular acceleration, , in terms of , , = Correct , and (but not ). Part B The force pulling the string is constant; therefore the magnitude of the angular acceleration of the wheel is constant for this configuration. Find the magnitude of the angular velocity of the wheel when the string has been pulled a distance . Note that there are two ways to find an expression for ; these expressions look very different but are equivalent. Express the angular velocity of the wheel in terms of the displacement , the magnitude of the applied force, and the moment of inertia of the wheel , if you've found such a solution. Otherwise, following the hints for this part should lead you to express the angular velocity of the wheel in terms of the displacement , the wheel's radius , and . = Correct This solution can be obtained from the equations of rotational motion and the equations of motion with constant acceleration. An alternate approach is to calculate the work done over the displacement by the force and equate this work to the increase in rotational kinetic energy of rotation of the wheel Part C Find , the speed of the string after it has been pulled by over a distance . Express the speed of the string in terms of , , , and = Correct ; do not include , , or in your answer. Note that this is the speed that an object of mass (which is less than ) would attain if pulled a distance by a force with constant magnitude . A Bar Suspended by Two Vertical Strings A rigid, uniform, horizontal bar of mass and length is supported by two identical massless strings. Both strings are vertical. String A is attached at a distance from the left end of the bar and is connected to the ceiling; string B is attached to the left end of the bar and is connected to the floor. A small block of mass is supported against gravity by the bar at a distance from the left end of the bar, as shown in the figure. Throughout this problem positive torque is that which spins an object counterclockwise. Use for the magnitude of the acceleration due to gravity. Harvaran Ghai Part A Find , the tension in string A. Express the tension in string A in terms of , = Correct Part B Find , the magnitude of the tension in string B. , , , , and . Which of the two identical strings will break first? string A string B Correct Part D If the mass of the block is too large and the block is too close to the left end of the bar (near string B) then the horizontal bar may become unstable (i.e. What is the smallest possible value of such that the bar remains stable (call it Express your answer for in terms of = Correct . and .Express the magnitude of the tension in string B in terms of = Correct . A Person Standing on a Leaning Ladder . . as computed in the previous part. Assuming that . and . the bar may no longer remain horizontal). . is not necessarily positive. and are held fixed. and . Part E Note that since . . what is the maximum block mass stable? In other words. . Part C If the bar and block are too heavy the strings may break. the bar will be for which the bar will always be ? . what is the maximum block mass such that Answer in terms of = Correct . )? . If stable no matter where the block of mass is placed on it. .. . . . Throughout the problem. .A uniform ladder with mass and length rests against a smooth wall. Under these circumstances. consider counterclockwise torques to be positive. A doityourself enthusiast of mass stands on the ladder a distance from the bottom (measured along the ladder). is the magnitude of the normal force exerted by the wall on the ladder. and . simplify your trig functions). Remember to pay attention to the relation of force . . Part B Suppose that the actual coefficent of friction is one and a half times as large as the value of the ladder? . The ladder makes an angle with the ground. That is. . None of your answers should involve (i. required between the ladder and the ground so that the . and is the magnitude of the normal force exerted by the ground on the ladder. There is no friction between the wall and the ladder. but there is a frictional force of magnitude between the floor and the ladder. . what is the magnitude of the force of friction that the floor applies to Express your answer in terms of and .e. and . = Correct . Harvaran Ghai Part A What is the minimum coeffecient of static friction ladder does not slip? Express = Correct in terms of . . Assume that the wheel has a moment of inertia about its rotation axis.A Rolling Hollow Sphere A hollow spherical shell with mass 1.324 Correct Weight and Wheel Consider a bicycle wheel that initially is not rotating. so that there is no slipping as it rolls down. =0. as a rectangular box might do on an inclined (frictionless) surface.80 . . Take the freefall acceleration to be = 9. A block of mass is attached to the wheel and is allowed to fall a distance . the shell would simply slide down the slope. rolls without slipping down a slope that makes an angle of 39. If there were no friction.80 the horizontal. Take the freefall acceleration to be = 9.80 .70 Correct Part B Find the magnitude of the frictional force acting on the spherical shell.44 Correct The frictional force keeps the spherical shell stuck to the surface of the slope. =3. Part C Find the minimum coefficient of friction needed to prevent the spherical shell from slipping as it rolls down the slope. =4.0 with Harvaran Ghai Part A Find the magnitude of the acceleration of the center of mass of the spherical shell. .Harvaran Ghai Part A Consider the case that the string tied to the block is attached to the outside of the wheel. in terms of . Find Express = Correct . = Correct . . at a radius . . Express in terms of . the angular speed of the wheel after the block has fallen a distance . for this case. for this case. and . Find . the angular speed of the wheel after the block has fallen a distance . Part C Which of the following describes the relationship between Correct and ? . Part B Now consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius . and . . . so there is no external torque being applied to the axis. and Harvaran Ghai . . The moment of inertia of the record is . There is friction between the two disks. Express = Correct Part B in terms of . ." the disks will eventually rotate with the same angular velocity. The axis of the disk is vertical and the disk is supported by frictionless bearings. After this "rotational collision. Record and Turntable Learning Goal: To understand how to use conservation of angular momentum to solve problems involving collisions of rotating bodies. Another disk (a record) is dropped onto the first such that it lands coaxially (the axes coincide).This is related to why gears are found on the inside rather than the outside of a wheel. The initial angular velocity of the second disk is zero. Part A What is the final angular velocity. of the two disks? . Consider a turntable to be a circular disk of moment of inertia rotating at a constant angular velocity around an axis through the center and perpendicular to the plane of the disk (the disk's "primary axis of symmetry"). The motor of the turntable is off. 0 days. the pressure force drops and the star undergoes a gravitational collapse until it becomes a neutron star. .87 During most of its lifetime. But after all the hydrogen "fuel" is consumed by nuclear fusion. No Some of the energy was converted into heat and sound as the frictional force. . deduce its radius. After undergoing gravitational collapse. 4. . stopping relative motion.Because of friction.100 . These "pulsing stars" were discovered in the 1960s and are called pulsars.46×104 m Correct Part B What is the speed of a point on the equator of the neutron star? Your answer will be somewhat too large because a star cannot be accurately modeled as a solid sphere. What is the final rotational kinetic energy. Harvaran Ghai Part A A star with the mass and size of our sun rotates once every 34. of the two spinning disks? Express the final kinetic energy in terms of . . one pulse per rotation. By treating the neutron star as a solid sphere. acting on the bottom disk due to friction with the record? Express the torque in terms of . 6. rotational kinetic energy is not conserved while the disks' surfaces slip over each other. = Correct of the twodisk system. . Neutron stars spin very rapidly and emit intense pulses of radio and light waves. = Correct . the star forms a pulsar that is observed by astronomers to emit radio pulses every 0. Part C Assume that the turntable deccelerated during time before reaching the final angular velocity ( is the time interval between the moment when the top disk is dropped and the time that the disks begin to spin at the same angular velocity). a star maintains an equilibrium size in which the inward force of gravity on each atom is balanced by an outward pressure force due to the heat of the nuclear reactions in the core. What was the average torque. In a neutron star. and Problem 13. and the initial kinetic energy angular velocities should appear in your answer. the electrons and protons of the atoms are squeezed together by gravity until they fuse into neutrons. torque acted.06×106 m/s Correct . Analyzing Simple Harmonic Motion This Error! Hyperlink reference not valid. shows two masses on springs, each accompanied by a graph of its position versus time. Harvaran Ghai Part A What is an expression for , the position of mass I as a function of time? Assume that position is measured in meters and time is measured in seconds. Express your answer as a function of . Express numerical constants to three significant figures. = Correct Part B What is , the position of mass II as a function of time? Assume that position is measured in meters and time is measured in seconds. Express your answer as a function of . Express numerical constants to three significant figures. = Correct Harmonic Oscillator Acceleration Learning Goal: To understand the application of the general harmonic equation to finding the acceleration of a spring oscillator as a function of time. One end of a spring with spring constant is attached to the wall. The other end is attached to a block of mass . The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined to be . The length of the relaxed spring is . The block is slowly pulled from its equilibrium position to some position the block is released with zero initial velocity. The goal of this problem is to determine the acceleration of the block and . along the x axis. At time as a function of time in terms of , , , It is known that a general solution for the position of a harmonic oscillator is , where , , and are constants. Your task, therefore, is to determine the values of , , and in terms of , connection between and ,and and then use the to find the acceleration. Harvaran Ghai Part A Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block time. Express your answer in terms of , = Correct , and the coordinate of the block as a function of . The negative sign in the answer is important: It indicates that the restoring force (the tension of the spring) is always directed opposite to the block's displacement. When the block is pulled to the right from the equilibrium position, the restoring force is pulling back, that is, to the leftand vice versa. Part B Using the fact that acceleration is the second derivative of position, find the acceleration of the block function of time. Express your answer in terms of , , and . as a = Correct Part C Find the angular frequency . Express your answer in terms of and = Correct . Note that the angular frequency and, therefore, the period of oscillations depend only on the intrinsic physical characteristics of the system ( and amplitude of the motion. ). Frequency and period do not depend on the initial conditions or the Energy of Harmonic Oscillators Learning Goal: To learn to apply the law of conservation of energy to the analysis of harmonic oscillators. Systems in simple harmonic motion, or harmonic oscillators, obey the law of conservation of energy just like all other systems do. Using energy considerations, one can analyze many aspects of motion of the oscillator. Such an analysis can be simplified if one assumes that mechanical energy is not dissipated. In other words, , where is the total mechanical energy of the system, is the kinetic energy, and is the potential energy. the following questions. . The kinetic energy of the system is. we will consider a horizontally moving block attached to a spring. are given. as shown in the figure Assume that the force constant . In this problem. the mass of the block. where is the mass of the block and is the speed of the block. and the amplitude of vibrations. We will also assume that there are no resistive forces. Answer Part A Which moment corresponds to the maximum potential energy of the system? A B . C. where is the force constant of the spring and is the distance from the equilibrium position. . B. For such a system. a common example of a harmonic oscillator is a mass attached to a spring. the potential energy is stored in the spring and is given by . Consider a harmonic oscillator at four different moments.Harvaran Ghai As you know. . labeled A. and D. Note that. . as always. that is. . it can be excluded from the calculations. since the gravitational potential energy is not changing in this case. In general. of . Therefore. moving toward equilibrium. . If the kinetic energy of the block is increasing. Correct Part D Which moment corresponds to the maximum kinetic energy of the system? A B . the mechanical energy of a harmonic oscillator equals its potential energy at the maximum or minimum displacement. at the equilibrium position. Part C Consider the block in the process of oscillating. . moving to the left. At that moment. the spring is stretched (or compressed) the most. Recall that . Therefore.C D Correct Part B Which moment corresponds to the minimum kinetic energy of the system? A B C D Correct When the block is displaced a distance from equilibrium. at the amplitude displacement. the block must be moving to the right. the maximum potential energy is course. and the block is momentarily at rest. moving away from equilibrium. Part F At which moment is A B ? ). .C D Correct Part E Which moment corresponds to the minimum potential energy of the system? A B C D Correct When the block is at the equilibrium position. Meanwhile. of course. Recalling what we found out before. Therefore. the block is at its maximum speed ( then be written as . The maximum kinetic energy can at the equilibrium position. Recall that and that . or . . we can now conclude that . At that moment. the spring is not stretched (or compressed) at all. . = Correct Energy of a Spring An object of mass attached to a spring of force constant oscillates with simple harmonic motion. The maximum displacement from equilibrium is and the total mechanical energy of the system is .C D Correct Part G Find the kinetic energy of the block at the moment labeled B. Express your answer in terms of and . Harvaran Ghai Part A What is the system's potential energy when its kinetic energy is equal to Correct Part B What is the object's velocity when its potential energy is ? ? . Harvaran Ghai of the fish? =4. . A fish hanging from the bottom of . The Harvaran Ghai Part A What is the value of the acceleration of gravity on this planet? Express your answer in meters per second per second. the block remains at rest when the spring is stretched a distance from its equilibrium length.42 Correct Vertical Mass-and-Spring Oscillator A block of mass is attached to the end of an ideal spring. Due to the weight of the block. what is the mass Express your answer in kilograms.55 The Fish Scale The scale of a spring balance reading from 0 to 205 has a length of 13. . The spring has an unknown spring . a space explorer constructs a simple pendulum of length 55.Correct Gravity on Another Planet After landing on an unfamiliar planet.0 explorer finds that the pendulum completes 108 full swing cycles in a time of 136 .7 Correct 136/108…2pi/ans…ans*0.95 Part A Ignoring the mass of the spring.5 the spring oscillates vertically at a frequency of 2. constant . =13. By measuring and (both fairly simple measurements). Find the resulting angular frequency of the block's oscillation about its equilibrium position. you can determine the value of the spring constant and the acceleration due to gravity experimentally. One way of thinking about this problem is to consider both and as unknowns. = Correct It may seem that this result for the frequency does not depend on either the mass of the block or the spring constant. . However. the magnitude of the acceleration due to gravity. = Correct Part B Suppose that the block gets bumped and undergoes a small vertical displacement.Harvaran Ghai Part A What is the spring constant ? Express the spring constant in terms of given quantities and . which might make little sense. and knowing the mass. these parameters are what would determine the extension of the spring when the block is hanging: . Express the frequency in terms of given quantities and . the magnitude of the acceleration due to gravity.