9/20/2014POTENTIAL ENERGY POTENTIAL ENERGY Due: 11:59pm on Wednesday, September 10, 2014 To understand how points are awarded, read the Grading Policy for this assignment. Item 1 Consider a spring, with spring constant k, one end of which is attached to a wall. The spring is initially unstretched, with the unconstrained end of the spring at position x = 0. Part A The spring is now compressed so that the unconstrained end moves from x W = ∫ xf xi = 0 to x . Using the work integral = L ⃗ F (x⃗ ) ⋅ dx⃗ , find the work done by the spring as it is compressed. Express the work done by the spring in terms of k and L. Hint 1. Spring force as a function of position The spring force vector F ⃗ as a function of displacement x from the spring's equilibrium position, is given by ⃗ ^ F = −kx i where k is the spring constant and ^i is a unit vector in the direction of the displacement of the spring (in this case, towards the right). Hint 2. Integrand of the work integral The work done by the spring is given by the integral of the dot product of the spring force and an infinitesimal displacement of the end of the spring: W = ∫ xf xi ⃗ F (x⃗ ) ⋅ dx⃗ = ∫ xf xi ⃗ F (x) ⋅ ^ i dx , ⃗ where the infinitesmal displacement vector dx⃗ has been written as ^i dx. Write F (x) in terms of given quantities, and then compute the dot product to find an expression for the integrand. (Note, ^i ⋅ ^i .) = 1 Express your answer in terms of k, x, and dx. ANSWER: ⃗ ^ F (x) ⋅ i dx = Hint 3. Upper limit of the work integral The lower limit of the work integral will be at xi . What will be the integral's upper limit? = 0 ANSWER: xf = ANSWER: http://session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 1/23 9/20/2014 W POTENTIAL ENERGY = − k 2 L 2 Correct Item 2 A hammer of mass M is moving at speed v 0 when it strikes a nail of negligible mass that is stuck in a wooden block. The hammer is observed to drive the nail a distance L deeper into the block. Part A Find the magnitude F of the force that the wooden block exerts on the nail, assuming that this force is independent of the depth of penetration of the − −− nail into the wood. You may also assume that v 0 ≫ √2gL, so that the change in the hammer's gravitational potential energy, as it drives the nail into the block, is insignificant. Express the magnitude of the force in terms of M , v 0 , and L . Hint 1. How to approach the problem One way to solve this problem is to use the work-energy theorem. To stop the hammer from moving, the wooden block-nail system must do a certain amount of work on the hammer. One expression for this amount of work involves F and the displacement of the hammer. In addition, the work-energy theorem implies that the initial kinetic energy of the hammer plus the work done on the hammer must equal the final kinetic energy of the hammer. This gives another expression for the work done that involves only the change in kinetic energy of the hammer. Equate the two expressions for the work done and solve for F . Hint 2. Find the work done in terms of F The work-energy theorem connects the work needed to stop the hammer with the change in the hammer's kinetic energy. Find the work W done on the hammer by the nail. Don't forget to consider the sign of your answer. Express your answer in terms of F and L. ANSWER: W = Hint 3. Find the change in kinetic energy of the hammer What is K f − Ki , the change in kinetic energy of the hammer? Express your answer in terms of M and v 0 . ANSWER: K f − Ki = ANSWER: 2 F = M (v 0 ) 2L Correct Part B Now evaluate the magnitude of the holding force of the wooden block on the nail by assuming that the force necessary to pull the nail out is the same as that needed to drive it in, which we just derived. Assume a relatively heavy M = 0.5 kg hammer (about 18 ounces), moving with speed v 0 = 10 m/s. (If such a hammer were swung this hard upward and released, it would rise 5 m). Take the penetration depth L to be 2 cm, which is appropriate for one hit on a relatively heavy construction nail. 1 lb = 4.45 N http://session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 2/23 How to start Draw a free-body force diagram showing all real forces acting on the block. Find the magnitude of the normal force What is the magnitude N of the normal force? Express your answer in terms of m . Find the magnitude of the friction force Write an expression for the magnitude Ff ric of the friction force. Hint 1.) lb = 4. L. A force of magnitude F is applied to pull the block up the incline at constant speed. Hint 2. θ .masteringphysics. g . g . ANSWER: N = ANSWER: F f ric = ANSWER: http://session. and F . g . Hint 1. m .com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 3/23 . Express your answer in terms of any or all of the variables μ . and θ . Part A What is the total work Wf ric done on the block by the force of friction as the block moves a distance L up the incline? Express the work done by friction in terms of any or all of the variables μ .9/20/2014 POTENTIAL ENERGY Express your answer to the nearest pound. m .45 N ANSWER: ⃗ |F | = 281 lb Correct Item 3 A block of weight mg sits on an inclined plane as shown. and θ . (Note: 1 . The coefficient of kinetic friction between the plane and the block is μ . g . m . g . L. ANSWER: WF = FL Correct Now the applied force is changed so that instead of pulling the block up the incline. ANSWER: Wf ric = −(μmgLcosθ) Correct Part D What is the total work WF done on the box by the appled force in this case? Express your answer in terms of any or all of the variables μ . L. θ . m .com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 4/23 . m . g . Part C What is the total work Wf ric done on the block by the force of friction as the block moves a distance L down the incline? Express your answer in terms of any or all of the variables μ . and F . and F . θ . L. ANSWER: WF = FL Correct Item 4 h http://session. and F .9/20/2014 POTENTIAL ENERGY Wf ric = −(μmgLcosθ) Correct Part B What is the total work WF done on the block by the applied force F ⃗ as the block moves a distance L up the incline? Express your answer in terms of any or all of the variables μ .masteringphysics. the force pulls the block down the incline at a constant speed. θ . U = mgh ANSWER: Incorrect. summarized as . kinetic energy of zero). The kinetic energy then will equal the change in potential energy from the initial point to the point just above the tabletop. Assume that the size of each bob is negligible. The maximum kinetic energy for a pendulum will occur when it is at the lowest point in its motion..9/20/2014 POTENTIAL ENERGY Six pendulums of various masses m are released from various heights h above a tabletop. correct answer withheld by instructor Part B Rank each pendulum on the basis of the maximum kinetic energy it attains after release. Hint 1. Part A Rank each pendulum on the basis of its initial gravitational potential energy (before being released) relative to the tabletop. Kinetic energy Each pendulum begins at rest (i. All the pendulums have the same length and are mounted such that at the vertical position their lowest points are the height of the tabletop and just do not strike the tabletop when released. as shown in the figures below. Gravitational potential energy Gravitational potential energy U is defined as the product of the mass of the object. Rank from largest to smallest To rank items as equivalent.masteringphysics. Rank from largest to smallest To rank items as equivalent. Hint 1. overlap them.e. ANSWER: http://session. all of the pendulums will be the same height above the tabletop. At this point. and the height of the object above a reference level. overlap them. the acceleration due to gravity.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 5/23 . ANSWER: http://session.masteringphysics. Since the kinetic energy is equal to the change in potential energy. and therefore reach the same velocity after falling equal distances.9/20/2014 POTENTIAL ENERGY Incorrect. although larger masses are acted upon by a larger gravitational force. a larger mass also has more inertia. overlap them. the final speed depends upon the change in height but is independent of mass. Thus. Notice that mass may be canceled from both sides.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 6/23 . All objects fall with the same acceleration. Hint 1. and a larger kinetic energy at its lowest point. a pendulum with larger mass has a larger potential energy upon release. correct answer withheld by instructor Part C Rank each pendulum on the basis of its maximum speed. Rank from largest to smallest To rank items as equivalent. The role of mass Both kinetic energy and gravitational potential energy are proportional to mass. The situation is similar to that of an object in free fall. In free fall. you may write down the equation mgΔh = 1 /2 mv 2 . Thus. These two effects cancel out. is its kinetic energy positive.masteringphysics. Determine the sign of the initial kinetic energy At the instant the ball leaves the thrower's hand. which we can denote by v 0 . The shape of the kinetic energy graph The ball's speed decreases linearly from its initial value. The velocity of the ball can be described by the equation . v(t) = v 0 − gt Since kinetic energy depends on the square of velocity. Part A Sketch a graph of the kinetic energy of the baseball. because of the constant acceleration due to gravity.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 7/23 . Hint 1.9/20/2014 POTENTIAL ENERGY Correct Item 5 A baseball is thrown directly upward at time t = 0 and is caught again at time t = 5 s. negative. or zero? ANSWER: positive negative zero Hint 2. how does the kinetic energy vary with time? Also. note that the ball reaches its maximum height halfway between the time that it leaves the thrower's hand and the moment it is caught. What is the speed of the ball when it reaches the maximum height? ANSWER: http://session. Assume that air resistance is so small that it can be ignored and that the zero point of gravitational potential energy is located at the position at which the ball leaves the thrower's hand. Initial gravitational potential energy The point at which the ball leaves the thrower's hand is defined to be the origin of the y axis. Hint 1. sketch a graph of the baseball's gravitational potential energy. The shape of the gravitational potential energy graph The potential energy of the ball is proportional to its height. and the height of the ball can be described by the equation h(t) = v0 t − 1 2 gt 2 . Hint 2. Using conservation of energy Since there are no nonconservative forces acting on the ball.masteringphysics.9/20/2014 POTENTIAL ENERGY Correct Part B Based on the graph of kinetic energy given (gray curve in the graphing window). Hint 3. Therefore. the total energy must remain the same throughout the motion. ANSWER: http://session. your graph of potential energy should be shaped such that potential energy plus kinetic energy does not change during the motion.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 8/23 . and the gravitational potential energy of the ball depends on the ball's height above the origin. Total energy The total energy of the baseball is the sum of its kinetic energy and gravitational potential energy. ANSWER: Correct Item 6 θ http://session.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 9/23 .masteringphysics. Hint 1.9/20/2014 POTENTIAL ENERGY Correct Part C Based on the kinetic and potential energy graphs given. sketch a graph of the baseball's total energy. Find the magnitude of the frictional force What is the magnitude f f of the frictional force? Express your answer in terms of μ .9/20/2014 POTENTIAL ENERGY A block of weight w sits on a plane inclined at an angle θ as shown. θ . and θ . Hint 1. http://session. Hint 2. A formula for work The work done by a constant force is given by the dot product of the force vector with the vector representing the displacement over which the force is applied. Compute the normal force Find the magnitude n of the normal force on the block. ANSWER: n = ANSWER: f f = ANSWER: Wf = −(μ⋅m⋅g⋅cosθ⋅L) Incorrect. Express your answer in terms of w and θ . L . The coefficient of kinetic friction between the plane and the block is μ . Hint 1. correct answer withheld by instructor Part B What is the work W done by the applied force of magnitude F ? Express your answer in terms of some or all of the following: μ .masteringphysics. A force F ⃗ is applied to push the block up the incline at constant speed. w . w . θ .com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 10/23 . L . Part A What is the work Wf done on the block by the force of friction as the block moves a distance L up the incline? Express your answer in terms of some or all of the following: μ . w . θ . L . L . correct answer withheld by instructor Part C What is the change in the potential energy of the block.9/20/2014 POTENTIAL ENERGY ANSWER: W = F ⋅L Incorrect. ANSWER: W = −(wsin(θ) − μwcos(θ))L Correct Part F W f http://session. w . as it moves a distance L down the incline? Express your answer in terms of some or all of the following: μ .masteringphysics. θ . ΔU . ΔU . ANSWER: ΔU = wLsinθ Correct Now the applied force is changed so that instead of pulling the block up the incline. Part D What is the change in potential energy of the block.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 11/23 . w . after it has been pushed a distance L up the incline? Express your answer in terms of some or all of the following: μ . ANSWER: ΔU = −wsin(θ)L Correct Part E What is the work W done by the applied force of magnitude F ? Express your answer in terms of some or all of the following: μ . L . w . the force F ⃗ pulls the block down the incline at a constant speed. θ . The tension in the cable is constant. and h is the height above the ground. Hint 1. Be certain that the method you are using will be accurate to two significant figures. that the work done on the elevator by the tension must equal the change in mechanical energy of the system. but since it lies somewhere between the grid lines. in fact. How to approach the problem In the previous part. Find the initial mechanical energy http://session. In this problem. Find the change in mechanical energy From the information given in the applet and the information found in Part A. because when the elevator stopped. so choose the graph that you can read most accurately. Hint 2. It shows an elevator with a small initial upward velocity being raised by a cable. you could use the graph of potential energy to determine the mass to two significant figures. Hint 2. The energy bar graphs are marked in intervals of 600 J. w . You would like to get the most accurate number you can. Express your answer in kilograms to two significant figures.9/20/2014 POTENTIAL ENERGY What is the work Wf done on the block by the frictional force? Express your answer in terms of some or all of the following: μ . L . ANSWER: Wf = −μwLcos(θ) Correct Item 7 Look at this applet.masteringphysics. The numerical data given in the window beneath the graphs do have two significant figures of accuracy. Express your answer in newtons to two significant figures. θ . and thus they could be used in combination with the data in the graph of the final energy to get a more accurate value for the work done on the elevator. Part A What is the mass m of the elevator? Use g 2 = 10 m/s for the magnitude of the acceleration of gravity. g is the = mgh magnitude of the gravitational acceleration. it is unlikely that you could determine the tension to the necessary accuracy. you could use the graph of work to find the tension. Hint 1. However. There may be more than one. where m is the mass of the object. Recall. Express your answer in joules to two significant figures. Needed formula Recall that the gravitational potential energy U near the earth's surface is given by U . the top of the potential energy bar lay right on one of the grid lines. determine the change in the total mechanical energy of the system ΔE . it is a good way to get an estimate with which to check your answer. Using the graphs Think about which graph(s) show energies that are directly related to the mass of the elevator. ANSWER: m = 60 kg Correct Part B Find the magnitude of the tension T in the cable. Hint 1.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 12/23 . Part A What horizontal force is necessary to hold the bag in the new position? Express your answer using two significant figures. at the instant when you run the simulation. Definition of mechanical energy Recall that the mechanical energy of a system is defined as the sum of kinetic energy and potential energy. E = K + U. A postal worker then displaces the bag to a position 2. Note that. and the information about the initial speed of the elevator given in the window beneath the bar graphs in the applet. how much work is done by the rope? Express your answer using two significant figures. ANSWER: F = 740 N Correct Part B As the bag is moved to this position. Thus.2m sideways from its original position. always keeping the rope taut. the total initial mechanical energy of the system is simply given by the initial kinetic energy of the elevator K = (1/2)mv 2 . the potential energy U of the elevator is zero. Hint 1.0m long.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 13/23 . which can be evaluated from the information about the mass of the elevator found in Part A. what is the initial mechanical energy E initial of the system? Express your answer in joules to two significant figures.masteringphysics. ANSWER: W = 0 J Correct http://session. ANSWER: = E initial J ANSWER: ΔE = J ANSWER: T = 480 N Correct Item 8 A 115kg mail bag hangs by a vertical rope 4.9/20/2014 POTENTIAL ENERGY Assuming that the potential energy of the elevator at the instant when you run the simulation is zero. If the spring is compressed a distance of 25. Hint 2. Correct Part B Find v m the muzzle velocity of the ball (i. and the ball never touches the inside of the gun. There is no air resistance. ANSWER: Wworker = 0 J Incorrect. Determine how to approach the problem http://session. = 0 Hint 1. correct answer withheld by instructor Item 9 A spring-loaded toy gun is used to shoot a ball of mass m = 1.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 14/23 .. how much work is done by the worker? Express your answer using two significant figures. force. Part A Which of the following statements are true? Check all that apply. Also. The spring has spring constant k = 667 N/m. Forces such as friction and drag are dissipative forces.0 centimeters from its equilibrium position y = 0 and then released. forces are those that always oppose the motion of the object on which they act.masteringphysics. No conservative forces act in this problem after the ball is released from the spring gun. Assume that all movement occurs in a straight line up and down along the y axis. the velocity of the ball at the spring's equilibrium position y ). ANSWER: Mechanical energy is conserved because no dissipative forces perform work on the ball. The force of tension in the spring is a conservative force. or nonconservative. Nonconservative forces Dissipative. or nondissipative. The forces of gravity and the spring have potential energies associated with them.e. the ball reaches a maximum height hmax (measured from the equilibrium position of the spring). the ball is always acted on by gravity. which is also a conservative.50 kg straight up in the air. Forces acting on the ball The ball is acted on by the spring force only when the two are in contact. as shown in the figure. Hint 1.9/20/2014 POTENTIAL ENERGY Part C As the bag is moved to this position. 0 cm and the final position is y . the location of the ball when the spring is compressed. Energy equations Recall that kinetic energy is given by the equation 1 2 mv 2 .9/20/2014 POTENTIAL ENERGY What physical relationship can you use to solve this problem? Choose the best answer. Choose from among the following three points: Check all that apply. . where k is the spring constant and x is the spring's displacement from equilibrium. y = 0 y = hmax .com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 15/23 . Hint 3. What kind(s) = 0 of energy does the system "spring-ball" have at the final position? http://session. Determine which two locations you should examine Pick the two points along the ball's path that would be most useful to compare in order to find the solution to this problem. the initial position is y = −25. What kind(s) = 0 of energy does the system "spring-ball" have at the initial position? ANSWER: kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials Hint 5. = 0 Hint 4. = 0 The elastic potential energy of a spring is given by 1 2 kx 2 .0 cm and the final position is y . . the equilibrium position of the spring. where v is the speed of the object and m is the object's mass. Find the initial energy of the system A useful statement of mechanical energy conservation relating the initial and final kinetic (K ) and potential (U ) energies is K initial + U initial = K f inal + U f inal In this situation. the maximum height that the ball reaches above the point y . the initial position is y = −25. Gravitational potential energy is given by mgy where y is the object's height measured from y . . . Determine the final energy A useful statement of mechanical energy conservation relating the initial and final kinetic (K ) and potential (U ) energies is K initial + U initial = K f inal + U f inal In this situation. which is the equilibrium position of the spring.masteringphysics. ANSWER: y = −25 cm . which is the equilibrium position of the spring. ANSWER: kinematics equations Newton's second law law of conservation of energy conservation of momentum Hint 2. Creating an equation From the hints you now know what kinds of energy are present at the initial and final positions chosen for the ball in this part of the problem. You also know that K initial + U initial = K f inal + U f inal. y = 0 y = hmax . the location of the ball when the spring is compressed. It has been determined that K initial is zero and U initial consists of two terms: gravitational potential energy and elastic potential energy. Choose from among the following three points: Check all that apply. Therefore. the maximum height that the ball reaches measured from y . ANSWER: vm = 4. ANSWER: y = −25 cm . = 0 Hint 2. In addition. Find the initial energy A useful statement of mechanical energy conservation is K inital + U initial = K f inal + U f inal Recall that in the problem statement. in meters.9/20/2014 POTENTIAL ENERGY ANSWER: kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials Hint 6.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 16/23 . .78 m/s Correct Part C Find the maximum height hmax of the ball. What kind(s) of energy does the ball have at the initial location? ANSWER: http://session. U f inal is zero. the equilibrium position of the spring. the initial and the final position should be taken as y = hmax .masteringphysics. in this situation. is set to correspond to the equilibrium position of the spring. Hint 1. Express your answer numerically. y location is at y = −25 cm = 0 . Choose two locations to examine Pick the two points along the ball's movement that would be most useful to compare in order to find a solution to this problem. Determine the final energy A useful statement of mechanical energy conservation is K initial + U initial = K f inal + U f inal In this situation. You know that K initial + U initial = K f inal + U f inal. K f inal is zero. ANSWER: v top = m/s ANSWER: kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials Hint 4.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 17/23 . the initial location is at y have at y = hmax = −25 cm . Find the speed of the ball at the top of its trajectory What is the speed v top of the ball at the top of its trajectory? Express your answer numerically.9/20/2014 POTENTIAL ENERGY kinetic only elastic potential only gravitational potential only kinetic and gravitational potential kinetic and elastic potential elastic and gravitational potentials Hint 3. http://session. It was determined that K initial is zero and that U initial consists of two terms: gravitational potential energy and elastic potential energy.17 m Correct In this problem you practiced applying the law of conservation of mechanical energy to a physical situation to find the muzzle velocity and the maximum height reached by the ball. Creating an equation From the above hints. and the final position should be taken as y = hmax . at which point it starts falling back to Earth. Hint 1. you now know what kind of energy is present at the inital and final positions chosen for the ball in this part of the problem. ANSWER: hmax = 1. In addition. What kind(s) of energy does the ball ? Hint 1. . Motion in the vertical direction Recall from kinematics that a ball travels upward until its speed decreases to zero. in meters per second. and v . Hint 1. would increase the maximum height reached by the ball? Check all that apply. ANSWER: reducing the spring constant k increasing the spring constant k decreasing the distance the spring is compressed increasing the distance the spring is compressed decreasing the mass of the ball increasing the mass of the ball tilting the spring gun so that it is at an angle θ < 90 degrees from the horizontal Correct Item 10 A force of 600N stretches a certain spring a distance of 0. it stops.00cm ? ANSWER: U2 = 1.60 J Correct Item 11 An object of mass m is traveling on a horizontal surface. stops.9/20/2014 POTENTIAL ENERGY Part D Which of the following actions.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 18/23 . you must choose the answer that explains why the object ultimately http://session.300m . and then recoils and travels in the opposite direction. The object has speed v when it reaches x = 0 and encounters a spring. When the object reaches x = 0 on its return trip.masteringphysics. Why does the object stop? Why does the object come to rest when it returns to x ? = 0 Although more than one answer may be true of the system. m . Part A What is the potential energy of the spring when it is stretched a distance of 0. the spring constant. The object compresses the spring. if done independently. There is a coefficient of kinetic friction μ between the object and the surface.0 J Correct Part B What is its potential energy when it is compressed a distance of 4. Express k in terms of μ . Part A Find k.300m ? ANSWER: U1 = 90. g . Hint 4. Hint 2. μ . Energy dissipated by friction is equal to (1/2)μgmt2 . μ is the coefficient of kinetic friction. m is the mass of the object. When the object reaches x = 0 the second time all of its initial energy has been dissipated by friction. μ . Energy stored in a spring The potential energy stored in a spring having constant k that is compressed a distance d is E spring = − ∫ F dx = ∫ d 0 kx dx = 1 2 kd 2 . How to approach this question Use the fact that E f inal = E initial + Wnonconservative to solve for the distance the spring was compressed. Therefore E f inal = 0 . so the only contribution to the system's energy comes from the kinetic energy of the object. Hint 1. Also the spring is not compressed. where μ is the coefficient of friction. Friction is a conservative force. Hint 3.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 19/23 . and d and g . ANSWER: E initial = Wnonconservative http://session.9/20/2014 POTENTIAL ENERGY comes to a stop. Check all that apply. = 0 the force of friction exactly balances the force exerted by the spring on the object. g is the magnitude of the acceleration due to gravity. Compute the compression of the spring By what distance d does the object compress the spring? Look at the initial condition when the object originally hits the spring and the final condition when the object returns to x . where m is the mass of an object. the object is not moving.masteringphysics. x = 0 At x is an equilibrium position and at this point the spring exerts no force on the object. = 0 Express d in terms of v . and t is the amount of time (since encountering the spring) the object has been moving. ANSWER: Work done by friction is equal to −mgμd. Hint 1. and d is the distance the object has traveled. the acceleration due to gravity. Hint 2. How does friction affect the system? Indicate which of the following statements regarding friction is/are true. ANSWER: When the object reaches x = 0 the second time all of its initial energy has gone into the compression and extension of the spring. How to approach this part Initially the spring is uncompressed. v . and g . Work done by friction is exactly equal to the negative of the energy dissipated by friction. g is the acceleration due to gravity. Find E initial What is the value of E initial ? Express your answer in terms of some or all of the variables m . The value of E f inal In its final position. Hint 3. How to approach this part The only nonconservative force in the system is the frictional force between the object and the surface it's on. d. However. Find Wnonconservative What is the value of Wnonconservative ? Express your answer in terms of some or all of the variables m . the acceleration due to gravity. Hint 7. If the object moves through a distance x. How to approach this part The only nonconservative force in the system is the frictional force between the object and the surface it's on. Hint 1. the object must be momentarily at rest. To obtain an equation involving k. v . and g .masteringphysics. http://session. so that you end up with an equation containing k. How to approach this part Since the spring is at its maximum compression. the work done by friction Wf riction is ⃗ Wf riction = f ⋅ s ⃗ = −μmgx ANSWER: Wnonconservative = ANSWER: d = Hint 5. The value of E f inal The value of E f inal is again zero.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 20/23 . So now E initial can be written in terms of k and other variables. but this time. Find E initial for this part of the motion What is the value of E initial for this part of the motion? Express your answer in terms of d and k.9/20/2014 POTENTIAL ENERGY Hint 4. at the two ends of the motion considered. Hint 1. so k was not part of the equation. Hint 6. If the object moves through a distance x. Hint 1. take the initial condition to be the moment when the spring is at its maximum compression and the final condition to be the moment when the spring returns to x = 0. Find Wnonconservative for this part of the motion What is the value of Wnonconservative for this part of the motion? Express your answer in terms of m . the acceleration due to gravity. Putting it all together In the previous part. you were able to find a relation for d in terms of the known quantities. So the only contribution to the energy is from the potential energy of the spring. the work done by friction Wf riction is ⃗ Wf riction = f ⋅ s ⃗ = −μmgx . μ . ANSWER: E initial = Hint 8. and d and g . the spring constant. use conservation of energy again. E f inal = E initial + Wnonconservative . the spring had no energy. μ . 20. find its speed when it returns to the bottom of the hill.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 21/23 . icy hill shown in the figure . or will it slide back down the hill? ANSWER: remain at rest at its highest point slide back down the hill Correct Part C If the rock does slide back down. ANSWER: h = 9.9/20/2014 POTENTIAL ENERGY ANSWER: Wnonconservative = ANSWER: k = 8m( μg v 2 ) Correct Item 12 A 28-kg rock approaches the foot of a hill with a speed of 15 m/s. The coefficients of static and kinetic friction between the hill and the rock are 0.0∘ above the horizontal. Express your answer using two significant figures. ANSWER: v = 12 m/s Correct Item 13 A 2.3 m Correct Part B Will the rock remain at rest at its highest point.masteringphysics. Express your answer using two significant figures. This hill slopes upward at a constant angle of 40. Part A Use energy conservation to find the maximum height above the foot of the hill reached by the rock. http://session.8-kg block slides over the smooth.75 and 0. respectively. The top of the hill is horizontal and 70 m higher than its base. 60kg is placed against a compressed spring at the bottom of a slope inclined at an angle of 34.05m up the incline from A. The mass of the spring is negligible. ANSWER: U1 = 110 J Correct Item 15 A sled with rider having a combined mass of 130kg travels over the perfectly smooth icy hill shown in the accompanying figure. The coefficient of kinetic friction between the block and incline is μ k = 0. a distance of 5. Part A How far does the sled land from the foot of the cliff? http://session.75m/s and is no longer in contact with the spring.80m/s2 . Part A Calculate the amount of potential energy that was initially stored in the spring.0∘ (point A).9/20/2014 POTENTIAL ENERGY Part A What minimum speed must the block have at the base of the hill so that it will not fall into the pit on the far side of the hill? Express your answer using two significant figures.45. the block is moving up the incline at a speed of 6.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 22/23 .masteringphysics. it projects the block up the incline. ANSWER: v min = 42 m/s Correct Item 14 A wooden block with mass 1. Take free fall acceleration to be 9. At point B . When the spring is released. http://session. The total mechanical energy of a system is constant only if conservative forces act. The total mechanical energy of a system is constant only if nonconservative forces act.76 out of a possible total of 20 points. is either all kinetic or all potential energy.masteringphysics. at any one instant.9/20/2014 POTENTIAL ENERGY ANSWER: d = 25. correct answer withheld by instructor Score Summary: Your score on this assignment is 109%. The total mechanical energy of a system is equally divided between kinetic and potential energy.5 m Correct Item 16 Part A Which of the following statements is/are true? Check all that apply. You received 21. Incorrect. Mechanical energy can be dissipated to nonmechanical forms of energy. ANSWER: The total mechanical energy of a system.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=2941502 23/23 .