264 CHAPTER 10 ROTATIONSample Problem Work, rotational kinetic energy, torque, disk Let the disk in Fig. 10-18 start from rest at time t 0 and Calculations: First, we relate the change in the kinetic also let the tension in the massless cord be 6.0 N and the an- energy of the disk to the net work W done on the disk, using gular acceleration of the disk be –24 rad/s2. What is its rota- the work–kinetic energy theorem of Eq. 10-52 (Kf Ki W). tional kinetic energy K at t 2.5 s? With K substituted for Kf and 0 for Ki, we get K Ki W 0 W W. (10-60) KEY IDEA Next we want to find the work W. We can relate W to We can find K with Eq. 10-34 (K 12 I2). We already know the torques acting on the disk with Eq. 10-53 or 10-54. The that I 12 MR 2, but we do not yet know v at t 2.5 s. only torque causing angular acceleration and doing work is : However, because the angular acceleration a has the con- the torque due to force T on the disk from the cord, which is stant value of 24 rad/s2, we can apply the equations for equal to TR. Because a is constant, this torque also must constant angular acceleration in Table 10-1. be constant. Thus, we can use Eq. 10-54 to write Calculations: Because we want v and know a and v0 ( 0), W t (uf ui) TR(uf ui). (10-61) we use Eq. 10-12: Because a is constant, we can use Eq. 10-13 to find v v0 at 0 at at. uf ui . With vi 0, we have Substituting v at and I 12 MR2 into Eq. 10-34, we find f i it 12t 2 0 12 t 2 12t 2. K 2 I 2 (2 MR )(t) 4 M(Rt) 1 2 1 1 2 2 1 2 Now we substitute this into Eq. 10-61 and then substitute the 1 2 4 (2.5 kg)[(0.20 m)(24 rad/s )(2.5 s)]2 result into Eq. 10-60. Inserting the given values T 6.0 N 90 J. (Answer) and a 24 rad/s2, we have K W TR(f i) TR(12t 2) 12TRt 2 KEY IDEA 12 (6.0 N)(0.20 m)(24 rad/s2)(2.5 s)2 We can also get this answer by finding the disk’s kinetic energy from the work done on the disk. 90 J. (Answer) Additional examples, video, and practice available at WileyPLUS Angular Position To describe the rotation of a rigid body about where u is positive for counterclockwise rotation and negative for a fixed axis, called the rotation axis, we assume a reference line is clockwise rotation. fixed in the body, perpendicular to that axis and rotating with the body.We measure the angular position u of this line relative to a fixed Angular Velocity and Speed If a body rotates through an direction.When u is measured in radians, angular displacement u in a time interval t, its average angular s velocity vavg is (radian measure), (10-1) r avg . (10-5) where s is the arc length of a circular path of radius r and angle u. t Radian measure is related to angle measure in revolutions and de- The (instantaneous) angular velocity v of the body is grees by 1 rev 360° 2p rad. (10-2) d . (10-6) dt Angular Displacement A body that rotates about a rotation Both vavg and v are vectors, with directions given by the right-hand axis, changing its angular position from u1 to u2, undergoes an angu- rule of Fig. 10-6. They are positive for counterclockwise rotation lar displacement and negative for clockwise rotation. The magnitude of the body’s u u2 u1, (10-4) angular velocity is the angular speed. We can describe h as being the distance the actual rotation axis has been shifted from the rotation axis through the com. (10-15) : the position vector r relative to the axis. (10-8) dt Both aavg and a are vectors. (10-7) for a body with continuously distributed mass. are Icom is the rotational inertia of the body about the axis through the v v0 at. and matic equations. PA R T 1 REVIEW & SUMMARY 265 Angular Acceleration If the angular velocity of a body for a system of discrete particles and defined as changes from v1 to v2 in a time interval t t2 t1. (10-14) Torque Torque is a turning or twisting action on a body about a : : rotation axis due to a force F . The r and ri in these t2 t1 t expressions represent the perpendicular distance from the axis of The (instantaneous) angular acceleration a of the body is rotation to each mass element in the body. given in Table 10-1. I Icom Mh2. 1 2 (10-16) . The Parallel-Axis Theorem The parallel-axis theorem relates the rotational inertia I of a body about any axis to that of the same body about a parallel axis through the center of mass: The Kinematic Equations for Constant Angular Accel- eration Constant angular acceleration (a constant) is an im. 1 2 (10-13) 2 20 2( 0). then the magnitude of the torque is 0 t 2 t . (10-36) portant special case of rotational motion. and the integration is car- d ried out over the entire body so as to include every mass element. (10-12) com. 0 0t 2 t . . the average angular acceleration aavg of the body is I 冕 r 2 dm (10-35) 2 1 avg . If F is exerted at a point given by 0 12 (0 )t. The appropriate kine- Here h is the perpendicular distance between the two axes. (10-45) at ar (radian measure). I is the ro- where a is the magnitude of the angular acceleration (in radians tational inertia of the particle or body about the rotation axis. (10-22) where tnet is the net torque acting on a particle or rigid body. rFt r⬜F rF sin . at a perpendicular distance r from the rotation axis. v vr (radian measure). 10-41. This line is called the line of action of F . tance between the rotation axis and an extended line running : : the point moves along an arc with length s given by through the F vector. (10-40. and a is per second-squared) of the body. the period T of equations used for translational motion and are 冕 the motion for the point and the body is f 2 r 2 W . A torque t The linear velocity : v of the point is tangent to the circle. If the body rotates through an angle u. the is positive if it tends to rotate a body at rest counterclockwise and point’s linear speed v is given by negative if it tends to rotate the body clockwise. v2 ar 2r (radian measure). (10-18) where v is the angular speed (in radians per second) of the body. The radial component of : a is the resulting angular acceleration about that axis. r is the moment arm of Ft. Similarly. 10-39) : Linear and Angular Variables Related A point in a rigid where Ft is the component of F perpendicular to : r and f is the : rotating body. The quantity r⬜ is the perpendicular dis- moves in a circle with radius r. : s ur (radian measure). angle between : r and F . The tangential component is tnet Ia. Newton’s Second Law in Angular Form The rotational The linear acceleration : a of the point has both tangential and analog of Newton’s second law is radial components. The SI unit of torque is the newton-meter (N m). (10-23) Work and Rotational Kinetic Energy The equations used r for calculating work and power in rotational motion correspond to If the point moves in uniform circular motion. where u is in radians. (10-17) and r⬜ is called the moment arm of F . (10-19. 10-20) i v dW and P . d (10-53) T (radian measure). (10-52) . (10-34) The form of the work – kinetic energy theorem used for rotating in which I is the rotational inertia of the body. 10-53 reduces to given by W t(uf ui). (10-55) Rotational Kinetic Energy and Rotational Inertia The dt kinetic energy K of a rigid body rotating about a fixed axis is When t is constant. (10-54) K 12I2 (radian measure).. Eq. defined as bodies is I 兺 mir 2i (10-33) K Kf Ki 12 I2f 12 2i W. : changing the magnitude of F 1. great. or maintain the magnitude of F F F F1 θ Denser Lighter Denser F2 Disk 1 Disk 2 Disk 3 Fig. either at the outer edge or at the interface of the two its center like a merry-go-round. greatest first. after. for four situations are: (a) 2 rad/s. Fig. decrease. c. ** View All Solutions Here ** . as shown. stants a. 10-24 Questions 7 and 8. Rank them according to the rotational inertia of the plate c t (s) around a perpendicular axis through them. made smaller. nega. 4 Figure 10-21b is a graph of the angular position of the rotating 8 Figure 10-24b shows an overhead view of a horizontal bar that : : disk of Fig. ω : : : F2? Do forces (b) F 1 and (c) F2 tend F4 gular velocity versus time for a disk to rotate the disk clockwise or coun. (a) 2 pivot. However. In each case. which is counterclockwise and at a con. or wise from u 0. The forces maintain the indicated materials. Question 9. with F2 at angle f to the bar. 5 rad/s. it is a (b) radial acceleration. 10-23. should we increase. or zero. should F2 be made larger. the rotation direction changes at a t 0 7 Figure 10-24a is an overhead view of a horizontal bar that can certain angular position uchange. 10-21a. gular position u versus time t for 90° Rank the forces according to the F2 three cases in which a disk is rotated 1 magnitude of the torque they create Fig. For a terclockwise? point on the disk rim. two forces F 1 and F2 act on a disk that turns about tially to the disk. Is the angular velocity of the disk positive. (b) 2 rad/s. b Three lettered points are indicated. negative. two horizontal forces act on the bar. Rank the following values of f accord- gular acceleration positive or negative? ing to the magnitude of the angular acceleration of the bar. and 110°. so as to change its angular velocity. greatest first. F1 P square that can rotate about point P. : tive. 10-25 1 2 3 est first. Each (a) (b) disk consists of the same two materials. F1 F2 F2 or at t 0 and (c) whether a is positive. the disk center. or whether it is at Question 2. (c) 2 rad/s. 10-22 Question 5. respectively. (density is mass per unit volume). b. and d according to the five forces of the same magnitude act magnitude of the (a) tangential and Fig. like a merry-go-round. 70°. 3 about point P. and (c) : stant rate. is rotated about the pivot point by two horizontal forces. Forces with identical magnitudes are applied tangen- 5 In Fig. Fig. and (d) 2 (a) (b) rad/s. 5 rad/s. left the same? u 0. (b) the rotational inertia about the disk center.266 CHAPTER 10 ROTATION ** View All Solutions Here ** 1 Figure 10-19 is a graph of the an. F 1 and F2. the denser material forms the outer half of the disk area. one denser than the other Fig. In disk 2. greatest first. greatest first: 90°. a θ (rad) Rotation axis 9 Figure 10-25 shows a uniform metal plate that had been square before 25% of it was snipped off. 5 rad/s. –90° bar is still not to turn. 10-21 Question 4. the torque due to the force. In disks 1 and 3. 10-20 (b) whether v is zero before. it forms the inner half : : of the disk area. For each case. (b) t 2 s. or zero at (a) t 1 s. θ at midlength along one of the edges. determine whether : angle between the bar and F2 is now decreased from 90° and the uchange is clockwise or counterclock. (a) To keep the angular speed con- stant. 2 Figure 10-20 shows plots of an. we are to decrease the angle u of F 1 without the angular acceleration of the disk. Rank the situations according to the work done by Fig. Rank the disks according to (a) the torque about angles during the rotation. Its initial and F1 final angular velocities. F3 on a strange merry-go-round. 5 rad/s. 10-19 Question 1. If the For each case. 10-22. and (c) t 3 s? (d) Is the an. rank the in. determine Fig. greatest first. 10-26 Question 10. F5 rotating like a merry-go-round. t a b c d 6 In the overhead view of Fig. Pivot point Pivot point φ 3 A force is applied to the rim of a disk that can rotate like a merry-go-round. 10-23 Question 6. but it is stationary. 10 Figure 10-26 shows three flat disks (of the same radius) that can rotate about their centers like merry-go-rounds. 5 ation of 1.0t2. where u is in radians and t is in seconds. During a certain 4. line turn in the positive direction? What are the (b) first and (c) (a) What minimum speed must the arrow have? (b) Does it matter second times the reference line will be at 12max? At what (d) where between the axle and rim of the wheel you aim? If so.60 rad/s. 60 revolutions later.0t4 4.0 rad. How many revolutions that time. it rotates 25 rad.5 m.0 s? (c) rev/s angular speed. and (c) the tangential acceleration of a spaceship rotate in coming to rest? taking a circular turn of radius 3220 km at a speed of 29 000 km/h? ** View All Solutions Here ** . (c) the time required to reach the 10 What are the angular velocities at (a) t 2. (e) Is its angular acceleration lar acceleration. •11 A disk. (b) What is its constant? angular acceleration? (c) How much time is required for it to com- plete the first 20 of the 40 revolutions? ••5 ILW A diver makes 2. first 2. 12. what are (a) the point’s angular position and (b) its angular ••13 ILW A flywheel turns through 40 rev as it slows from an velocity? (c) What is its angular velocity at t 4. a flywheel has an angular velocity of 4.0 s interval? eight equally spaced spokes and a radius of 30 cm. and (d) the number of revolutions from rest What is the average angular acceleration for the time interval that until the time the disk reaches the 10 rev/s angular speed. it turns through an angle of 120 rad. How long does it take to rotate through (a) the rev/s. During plate at 85 mi/h with a spin of 1800 rev/min. Assuming that the wheel started from rest.wiley. (b) the ra- (a) how much time does it take and (b) through what angle does it dial acceleration. at u0 0. what is the (rad/s) and (b) the angular position (rad) as functions of time (s). angular velocity of the disk at the end of the 5. At time t 0. find the time for it to come to rest.0 rad/s2. 10-4 Rotation with Constant Angular Acceleration as seen by (a) the pilot and (b) an observer on the ground? The •9 A drum rotates around its central axis at an angular velocity of plane’s velocity is parallel to the propeller’s axis of rotation. (a) Through what maximum angle umax will the reference arrow and the spokes are very thin.0t 2 2.5 revolutions on the way from a 10-m- high platform to the water.0 s? (d) Calculate angular speed of 1. the sec. (b) the time required to given by u 4.0 4. what are the magnitudes of (a) the angular acceleration does the baseball make on its way to home plate? For simplicity. ••6 The angular position of a point on the rim of a rotating wheel is Calculate (a) the angular acceleration. at radius 1. (a) over the edge of a counter.5 rad? (f) Graph u versus t.0t 3. a through the wheel without hitting constant angular acceleration of 0.0 s and ends at t 4.25 rad/s2. and a reference line any of the spokes.0 s. (a) Assuming a constant angu- its angular acceleration at t 2. ••14 A disk rotates about its central axis starting from rest and find the average angular velocity during the dive.0 s? watch? Answer in radians per second. where u is in radians and t is in seconds. and (c) the hour hand of a smoothly running analog the disk turn during the next 5. what are the (a) gle does the disk rotate during that time? smallest and (b) largest angular speeds that cause the toast to hit •12 The angular speed of an automobile engine is increased at a and then topple to be butter-side down? constant rate from 1200 rev/min to 3000 rev/min in 12 s. begins at t 2.0t 3. (b) the gular acceleration unchanged.0 s. 10-2 The Rotational Variables •10 Starting from rest. it rotates as it falls. initially rotating at 120 rad/s. what negative time and (e) positive time will the reference line be at is the best location? 10.0t 2 t 3.0 s? What are the instanta- neous angular accelerations at (d) the beginning and (e) the end of ••15 SSM A wheel has a constant angular acceleration of this time interval? 3.0 s and (b) t 4. a disk rotates about its central axis with •1 A good baseball pitcher can throw a baseball toward home constant angular acceleration. a in radians per second-squared and t in seconds.7 rad/s. with through (e) on the graph.50 rad/s2. and indicate the answers to (a) •••8 The angular acceleration of a wheel is a 6. In 5. its angular speed is 15 rev/s.0 rad/s2. and (b) the average angular velocity? (c) What is the instantaneous assume that the 60 ft path is a straight line. sec. complete the 60 revolutions. accelerates with constant angular acceleration. It is mounted on a ••16 A merry-go-round rotates from rest with an angular acceler- fixed axle and is spinning at 2.5 rad/s to a stop.com/college/halliday • – ••• Number of dots indicates level of problem difficulty ILW Interactive solution is at Additional information available in The Flying Circus of Physics and at flyingcircusofphysics. Write expressions for (a) the angular velocity plane flies at a speed of 480 km/h relative to the ground. 10-5 Relating the Linear and Angular Variables wheel has an angular velocity of 2.0 s interval. You want to shoot a 20-cm. (a) What is ••4 The angular position of a point on a rotating wheel is given by its angular acceleration in revolutions per minute-squared? (b) How u 2. Assume that the Fig. how long •••7 The wheel in Fig.00 rev and (b) the next 2. linear speed of a point on the tip of the propeller. •19 What are the magnitudes of (a) the angular velocity. PA R T 1 PROBLEMS 267 ** View All Solutions Here ** Tutoring problem available (at instructor’s discretion) in WileyPLUS and WebAssign SSM Worked-out solution available in Student Solutions Manual WWW Worked-out solution is at http://www. At many revolutions does the engine make during this 12 s interval? t 0.0 s? (d) With the an- •2 What is the angular speed of (a) the second hand.20 rad/s2. 10-27 Problem 7. If the distance to the How much time does the disk take to stop? (b) Through what an- floor is 76 cm and for rotation less than 1 rev.com sec. Assuming zero initial vertical velocity.0 rad/s and an angular posi- •18 If an airplane propeller rotates at 2000 rev/min while the air- tion of 1. through what additional angle will minute hand. 10-27 has has it been in motion at the start of this 4.00 rev? long arrow parallel to this axle and ••17 At t 0. At one time it is ro- tating at 10 rev/s. is slowed down ••3 When a slice of buttered toast is accidentally pushed with a constant angular acceleration of magnitude 4. If the drum then slows at a constant rate of 4. 0 s? Light Mirror •24 A vinyl record is played by rotating the record so that an ap. gaining angu- turning at 75 rev/min. causing it to oscillate. (a) value of a1? Calculate the acceleration of the seed. travels to a •21 Between 1911 and 1990. (a) What is the con. ••25 SSM (a) What is the angular speed v about the polar axis of a point on Earth’s surface at latitude 40° N? (Earth rotates about that axis. assuming the belt does not slip. in revolutions per minute-squared. A beam of light passes through one of component of acceleration? the slots at the outside edge of the wheel. 10-29 Problem 29. The equipment converts those oscillations to electrical signals and then to sound. what are the magnitudes of ••29 An early method of measuring the speed of light makes use the point’s (a) tangential component of acceleration and (b) radial of a rotating slotted wheel. 10-28.033 s that is increasing at the 0. The centrifuge 3. 10-29. how many years from now will the pulsar acceleration period. (Hint: If the radians and t is in seconds. A watermelon time required for the rotation? (b) What is the corresponding seed is on the turntable 6. as in Fig. The magnitude of the centripetal acceleration of any por- the particle in (c)? tion of the disk is not to exceed 400 m/s2. with a radius of 0. When t 5.0 1. Measurements taken when the average angular speed of the tower’s top about its base? mirror is L 500 m from the wheel indicate a speed of light of •22 An astronaut is being tested in a centrifuge.20 m is rotating L at an angular speed of 200 rev/min. The tower is 55 m tall. Suppose that a record turns at the rate of 3313 rev/min. of what distance does a point on the rim move during the spin-up? the wheel during the slowdown? (b) How many revolutions does ••31 A disk. dius rA 10 cm is coupled by belt C B to wheel C of radius rC 25 cm.2 mm/y. (a) What ••26 The flywheel of a steam engine runs with a constant angular is the tangential acceleration of a point on the rim of the flywheel velocity of 150 rev/min. source perpendicular proximately circular groove in the vinyl slides under a stylus.0 cm.30t2.75 mm. rA rC stop rotating? (c) The pulsar originated in a supernova explosion ••28 In Fig. At t 0.0 105 km/s. netic energy of 24 400 J when rotating at 602 rev/min. One such slotted wheel has a radius of 5.) (b) What is the linear speed v of the point? What are (c) ••30 A gyroscope flywheel of radius 2. sec.25 m. Consider a point on the object that is 4. is to be rotated like a the wheel make before stopping? (c) At the instant the flywheel is merry-go-round through 800 rad.2 rad/s2 until its angular speed is 2760 rev/min. 10-28 Problem 28. 10-6 Kinetic Energy of Rotation B The angular speed of wheel A is in.268 CHAPTER 10 ROTATION ** View All Solutions Here ** •20 An object rotates about a fixed axis. In radians per second. the linear speeds at the two rims must be equal.0 belt does not slip.0 cm from the axis of rotation. (b) What is the minimum value of the coefficient of static fric- beam the way a lighthouse emits a light beam.) cm from the axis of rotation. (a) What is the least ••27 A record turntable is rotating at 3313 rev/min. When steam is shut off. At what rate (hits per second) do the bumps hit the stylus? Fig. this point when the flywheel is spinning at full speed? (c) Through stant angular acceleration. and returns to the wheel just in time to pass through tower at Pisa. find the initial T. (a) What is the angular speed of the flywheel in radians per second? (b) What is the linear speed of a point on the rim of the flywheel? (c) What constant angular ac- celeration (in revolutions per minute-squared) will increase the Light wheel’s angular speed to 1000 rev/min in 60.83 cm is accelerated from v and (d) v for a point at the equator? rest at 14. (c) tangential acceleration. The pulsar in the Crab neb- ing from rest and undergoing a constant angular acceleration for ula has a period of rotation of T 0. wheel? (b) What is the linear speed of a point on the edge of the where t is in seconds and u is in radians. (a) What is the pulsar’s angular acceleration for the seed not to slip during the a? (b) If a is constant. the top of the leaning bell distant mirror. what are wheel? the magnitudes of the astronaut’s (a) angular velocity. The period T of rotation is found (c) Suppose that the turntable achieves its angular speed by start- by measuring the time between pulses.0 s? (d) How many beam revolutions does the wheel make during that 60.40e2t. where u is in speed of 100 rev/min. what is the tangential component of the lin- lar speed at the constant rate a1 through the first 400 rad and ear acceleration of a flywheel particle that is 50 cm from the axis of then losing angular speed at the constant rate a1 until it is again rotation? (d) What is the magnitude of the net linear acceleration of at rest. and (d) radial acceleration? •23 SSM WWW A flywheel with a diameter of 1. what is the cm and 500 slots around its edge.6 rad/s2. the Rotating groove being played is at a radius of 10. Assuming constant a. (a) What is the (constant) angular speed of the has a radius of 10 m and.26 105 s/y. assuming that it does not •••32 A pulsar is a rapidly rotating neutron star that emits a radio slip.2 h. wheel A of ra- A seen in the year 1054. ** View All Solutions Here ** . starting from rest. We receive a radio tion between the seed and the turntable if the seed is not to slip? pulse for each rotation of the star. to light beam Bumps in the groove run into the stylus. moved toward the south at an average rate of the next slot in the wheel. Calculate the minimum coefficient of static friction required rate of 1.0 s. Find the time needed for wheel C to reach an angular of a reference line on the object is given byu 0.25 s. (b) linear velocity. and the bumps in the slotted wheel groove are uniformly separated by 1. •33 SSM Calculate the rotational inertia of a wheel that has a ki- creased from rest at a constant rate Fig. in starting. the friction of the during this spin-up process? (b) What is the radial acceleration of bearings and of the air stops the wheel in 2. rotates according to u 0. and the angular position of 1. Italy. of radius 0. which form the sides of the square. The scale on the v axis is set by (a) the innermost one and (b) the outermost one? s 6. (a) What is the kinetic energy ◊ (rad/s) of the flywheel after charging? (b) If the truck uses an average ωs power of 8. (b) y. with c mass 0. by what percentage does the rotational inertia of the assembly lar to the large faces. If we remove one particle (that is. What is the rotational Axis inertia of this rigid body about an axis that (a) passes through the O m m m midpoints of opposite sides and lies in the plane of the square. and (c) lies in the plane of the square Fig.2 kg. 10-33 Problem 40.0 cm.0 m. 10-35 has mass 0. 33% of the mass). uniform cylinder with a mass of 500 kg and a radius of 1. and (c) z (a) (b) axes? (d) Suppose the answers to (a) and (b) are A and B.0 m 2. L tral (longitudinal) axis at 235 rad/s. 25 g. y 2. What are the rota- h (m) tional inertias of this collection about the (a) x. (a) What is the magnitude of the rod’s angular ac. What is the rotational kinetic energy of (a) the O smaller cylinder. Rotation Fig. 10-34. have the same mass of 1. The tened to each other. 10-30 Problem 34.172 kg and edge lengths a 3. 10-34 Problem 41. Measured about O.0 kW. by two thin rods.0 rad/s.0 cm. about an axis perpendicular to the stick and located ••43 SSM WWW The uniform solid at the 20 cm mark. Figure 10-31b gives ••41 In Fig. 25 g. PA R T 1 PROBLEMS 269 ** View All Solutions Here ** •34 Figure 10-30 gives angular speed versus time for a thin rod around the rotation axis decrease when that removed particle is that rotates around one end.0 mg.5 cm.0 m square and held there by four mass- L less rods. b 8. x 3.6 cm and mass M M 1. what are the I (kg • m2) h combination’s (a) rotational inertia and (b) kinetic energy? ••42 The masses and coordinates of four particles are as follows: 50 g. 30 g.0000 m and (total) mass M 100. the rotational inertia I of the disk about the axis as a function of each with mass m 0.0100 kg particles that have been cm. are fas. x 0.150 kg m2.1 0. 10-31 Problem 36.50 kg each are placed at the vertices of a 2. what percentage error would we make in us- ing the formula in Table 10-2e to calculate the rotational inertia? sec. x 2.30 rad/s. for how many minutes can it operate between chargings? ••40 Figure 10-33 shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length L 1. x 2.4 cm.25 m. two particles. mass M and length L. ω m d that distance h.56 kg.60 J. and c 1. y 3. ••39 Trucks can be run on energy stored in a rotating flywheel. •36 Figure 10-31a shows a disk that can rotate about an axis at a radial distance h from the center of the disk. and (b) the larger cylinder. One such flywheel is a solid.0 cm. of radius 0. 10-7 Calculating the Rotational Inertia •35 SSM Two uniform solid cylinders. from the center out to the edge of the disk.4 a •38 Figure 10-32 shows three 0.0 s. respectively. each rotating about its cen. 10-32 Problems 38 and 62. the rod has a rotational kinetic energy with an electric motor getting the flywheel up to its top speed of of 1. Calculate its ro- glued to a rod of length L 6.050 kg m2 and IB 0.85 kg. each d m What is the mass of the disk? with length d 5.0 cm.0 cm. celeration? (b) At t 4. (b) d d d passes through the midpoint of one of the sides and is perpendicu- lar to the plane of the square. Then what is the answer to (c) in axis terms of A and B? b •37 SSM Calculate the rotational inertia of a meter stick.00 cm and negligible mass.75 m? Fig. tion axis at O. sembly can rotate around a perpendicular axis through point O at Fig.) block in Fig. M scale on the I axis is set by IA 0. ••44 Four identical particles of mass 0. The combination ro- Rotation axis Axis IB tates around the rotation axis with O the angular speed v 0.2 cm. (Treat the stick as a thin rod.0 IA 0 0. (a) 0 1 2 3 4 5 6 What is the rotational inertia of the arrangement about that axis? (b) If we approximated the arrangement as being a uniform rod of Fig.0 cm. The disk arrangement can rotate 0 t (s) about a perpendicular axis through its central disk at point O. y 4. Fig. 10-35 through one corner and perpendicu- the left end. and to a rota. What is its kinetic energy at t 0? 200p rad/s. The as- tational inertia about an axis Problem 43.25 kg but differ in radius. y 4. and passes through two diagonally opposite particles? ** View All Solutions Here ** . u1 75.m1 m2 Axle sity (mass per unit volume) is to be rotated ation of the blocks? What are (b) ten- Fig.100 N. Her rotational inertia about her center of mass is 12. FB 16 N at B.0 cm aligned with point O (Fig. The h 220 ms. 10-40 Problem 53. 10-39.20 rad/s in with his left foot swept out. R his initial angular acceleration : and the pulley. A string is String and R 12 cm. Fg speed about her center of mass changes from zero to 6.30 m.25 s the disk has an angular velocity of 250 rad/s counterclockwise. your oppo- nitude of the torque about the pedal arm’s pivot when the arm is at nent rotates around his right foot angle (a) 30°. time t 0. 10-42b).0 rotational axis is through point O. 10-36 is pivoted at O. block 1 has mass kg m2. gi is (a) negligible and (b) horizontal with a magnitude of 300 N and has radius R 5. As a result.0 m F3 from O. If r1 1. 10-9 Newton’s Second Law for Rotation shows a simplified diagram of •49 SSM ILW During the launch from a board. what is the wheel’s rotational inertia? d tional inertia about point O is 65 ••51 In Fig. and his rota- 25. plate face and through point O. During the launch. so that at time t 1. which is a horizontal dis- tance d 28 cm from point O.) string is pulled for 5. 10-42a. block 2 falls 75. 10-41 Problem 54.00 s. 10-36 Problem 45.90 N. and F4 5. the forces a string is wrapped around a top. and FC 19 N at C. O •50 If a 32. axis •46 The body in Fig.270 CHAPTER 10 ROTATION ** View All Solutions Here ** sec. a cylinder having a mass of 2. : kg m2.0°. r1 r 2 F2 and u2 60. Three forces act on it: FA 10 N at point A.500 kg and radius 2. F2 4. O (a) What is the magnitude of the acceler. 10-39 Problem 52. what is the net F1 O O torque about the pivot? Rotation r Fig. What is the magnitude of Fig.0 m from O. and (c) 180° with the vertical? and onto the mat. plied tangentially to the rim as indicated.00 cm. and a downward pulling on his gi (uniform) toward force of 111 N is applied to the pedal by the rider. F3 2. what is ••54 In a judo foot-sweep the magnitude of the gravitational torque calculated about the pivot? move.0 N. Fa that side. the disk Problem 55. The rota. The gravitational force Fg on him age angular acceleration and (b) the average external torque on effectively acts at his center of her from the board? mass.4 m? from rest.15 m. Forces are applied as shown: cm is glued to the plate. 10-38 around an axle that is perpendicular to the sion T2 and (c) tension T1? (d) What is Problems 51 and 83. Find the (a) magnitude and (b) direction of the an. Figure 10-41 com sec. m1 460 g. block 2 has mass m2 500 g.0 N m torque on a wheel causes angular acceleration His mass is 70 kg. F1 θ2 θ1 F4 R 4. What is the mag. you sweep your opponent’s left foot out from under him while •48 The length of a bicycle pedal arm is 0. wrapped around the edge of the disk the way (b) gular acceleration of the cylinder.0 kg can rotate cular disk of mass 0. ** View All Solutions Here ** .00 cm and a mass of 20. an irregularly shaped Plate without the cord slipping on the pulley. r 5. F2 4. 10-37 Problem 46. The disk has a radius of B 2.0°.20 N. Starting at Fig. •47 SSM A small ball of mass 0.0 N. and two F1 forces act on it as shown.25-m-long massless rod. 10-42 maintain their same angles relative to the cylinder. 4. 10-8 Torque •45 SSM ILW The body in Fig. As a result. which is mounted on a about point O if your pull Fa on his horizontal axle with negligible friction. What is magnitude F2? 1. 10-38.0 grams and is initially at rest. a diver’s angular your opponent as you face him. 3. plastic plate with uniform thickness and den.152 m.0 cm in 5.75 kg is attached to one end of a : Force F1 has a magnitude of 0.0 m F2 from O. FC 135° C A ••53 Figure 10-40 shows a 160° uniform disk that can rotate O F1 90° around its center like a merry-go- FB F2 round. Then the Fig. When the resulting pendulum is 30° from the vertical.0 rad/s2. r2 2. A cir. (During the rotation. What is the net torque about O? FA Fig. what are the magnitudes of (a) her aver. 8. (b) 90°.00 about its central axis through point O.0 N. When released applied at height h 1. two forces are to be ap- Fig. and the other end of the rod is hung from a pivot. Also. with its center F1 6. (a) the magnitude of the pulley’s angular ac- tional inertia of the plate about that axle is celeration? (e) What is its rotational inertia? measured with the following method.00 s T1 T2 ••55 In Fig. 10-37 is pivoted at O.0 N. Disk ••52 In Fig. 0 103 kg m2 about its the pulley. three 0. The rod is held horizontally on the fulcrum over a pulley of rotational inertia I 3. with F in newtons and t in seconds.5 cm can rotate about a vertical axis on frictionless bearings length L1 L2. 10-45 Problem 67.0100 kg particles have been glued to a the energy given to the assembly in rod of length L 6. What is the rotational inertia of the plate stant it makes an angle of 35. The pulley is initially at rest. The disks both passes through its lowest position? have a uniform density (mass per unit Fig.5 s? (b) What is the just before it hits the floor. The resulting an.50t 0.) values of (c) t and (d) F for the larger sphere? ••64 A uniform cylinder of radius 10 cm and mass 20 kg is 69 In Fig.0 cm. of a larger disk of radius R 4.6 kW) when rotating at a m speed of 1800 rev/min. (a) What is the ••63 SSM ILW A meter stick is held vertically with one end on magnitude t of the torque required to bring the smaller sphere the floor and is then allowed to fall.00 cm so (a) What is the rotational inertia of the cylinder about the axis of that the disks lie in the same plane.30t2.75 m and mass 0.0 rad/s. (Hint: Consider the stick to be a thin rod and use the con. ruptured.) (c) particles 1 and 2. It must be brought to a stop in 15. but energy (in joules) versus the square of its rotation rate (in radians.0° with the vertical as it falls. 10-46 Problem 69. (a) around a horizontal axis in the plane Rotation How much work must be done to stop it? (b) What is the required axis of the rod and hoop. attached to the ends 1 2 •••66 A uniform spherical shell of mass M 4.226 m and the other has a radius of 0. what would be the assembly’s angular around a perpendicular axis through point O at one end.0 m.0 rad/s? (d) What is the slope of a plot of the assembly’s kinetic 68 Two uniform solid spheres have the same mass of 1. Find the speed of the other end from rest to an angular speed of 317 rad/s in 15. cm and L2 80 cm.400 N that is applied •••65 A tall. and is attached to a small object of mass m 0. At t 3. not a torque. A massless cord passes around the equator of the shell. 10-32.20 m. one has a radius of 0. and (b) the tangential acceler- ••56 Figure 10-43 shows L1 L2 ation of the top.00R). 10-18. it will rotate rotating at 280 rev/min. The force magnitude varies in time as F 0. at its rim. How speed about the rotation axis when it passes through the upside- much work is required to change the rotational rate (a) from 0 to down (inverted) orientation? 20. 10-10 Work and Rotational Kinetic Energy I.0 s. ** View All Solutions Here ** .Treat the chimney as a thin rod of length 55.0 103 kg m2 and radius and then released. The assembly is upright.854 m. each of At what angle u is the tangential acceleration equal to g? mass m. (b) from 20. (b) Repeat (a) with R 5. 2. a small disk of radius mounted so as to rotate freely about a horizontal axis that is paral. •59 An automobile crankshaft transfers energy from the engine to the axle at the rate of 100 hp ( 74. 10-44).0 kg wheel. squared per second-squared)? Each can rotate about an axis through its center. cylindrical chimney falls over when its base is by the string tangentially to the edge of the disk. It is pulled to one side and then allowed to swing •••67 Figure 10-45 shows a rigid like a pendulum.65 kg.150 m) and a thin rod’s kinetic energy at its lowest position and (b) how far above that radial rod (of mass m and length L position the center of mass rises. R sec.0 rad/s. Assuming that Fig. 10-43 Problem 56. What is the speed of the object when it has fallen 82 cm axle and a radius of 10 cm. the cord does not slip on •••57 A pulley.0 rad/s to Additional Problems 60. assuming that the end on the floor does magnitude F of the force that must be applied tangentially at the not slip. is Rod we give it a slight nudge. M 400 g. r 5. 10-46. R 8. and m 50 g in Fig.0 cm from the central longitudinal axis of the cylinder.60 kg. r •58 (a) If R 12 cm. through the average power? lower end of the rod. What torque (in newton-meters) does the crankshaft deliver? Fig.At the in- gular speed is 114 rad/s. sphere’s equator to give that torque? What are the corresponding servation of energy principle. r 2. find the speed of the block after it has descended 50 cm starting from rest.42 kg is suspended freely from one end.0 cm. find (a) the Hoop and radius R 0. passing through its lowest position with angular assembly of a thin hoop (of mass m speed 4. What are the magnitudes of the initial accelera.0 rad/s.5 kg and radius of a rigid massless rod of Fig.00 cm and negligible mass and can rotate such a nudge is negligible. 10-44 Problem 66. with L1 20 (Fig. (Hint: Use energy considerations. tions of (a) particle 1 and (b) particle 2? There is no friction on the pulley’s axle. ••62 In Fig.00 cm has been glued to the edge lel to and 5. •60 A thin rod of length 0. what are about the axle? (a) the radial acceleration of the top. is acted on by a force applied tangentially after being released from rest? Use energy considerations.0 rad/s to 40. Solve the problem using energy conservation principles. PA R T 1 PROBLEMS 271 ** View All Solutions Here ** and plate are rotated by a constant force of 0. what is the angular speed of the cylinder as it ter of the larger disk. The O rotation? (b) If the cylinder is released from rest with its central disks can be rotated around a perpen- longitudinal axis at the same height as the axis about which the dicular axis through point O at the cen- cylinder rotates. but if •61 A 32.0 s what are its (a) angular acceleration and (b) angular speed? M. and (c) from 40. essentially a thin hoop with radius 1. with a rotational inertia of 1. Neglecting friction and air resistance. 20 kg T2 blocks are connected by a massless 76 Starting from rest at t 0. Here ** pole to help: If he leans.) (c) How angular velocity at u2 is 15. the pulley turns through 0. and is attached to the rotor axle by a single bolt. Why?) The radius k of the equivalent hoop is called the radius of gyration (b) Calculate the torque that must be applied to the rotor to bring of the given body.0 kg pole he carries has its com 10 cm to the left of the wire? 71 SSM In Fig. an astound- of mass over the wire (or rope). Because of of the H.06 kg. (a) At what time t will passes through the horizontal position. (b) the mag. (b) the body when the plane of the H is vertical? the retarding torque.00 s interval? ries no pole and (b) the 14. What is the angular speed of torque due to friction.0 m and can pivot about a horizontal.0 rad/s.Fig. when it 104 kg m2.0 rad in 6. (a) What was its angular velocity much work does the torque do on the blade in order for the blade at u1? (b) What is the angular acceleration? (c) At what angular po- to reach a speed of 320 rev/min? sition was the disk initially at rest? (d) Graph u versus time t and an- 74 Racing disks. (a) Find its (con- pulley’s axis is frictionless.00 s. and (d) can still be computed without additional infor. has a mass of and a radius k given by 110 kg. inertia of a thin hoop of mass M and radius R/ 22 about its central tarding torque is known not to be constant. 0 that the reference lines are momentarily aligned? Polytechnic Institute. Why position u1 10.0 rad to angular position u2 70. and stops in 30 s after the motor is turned off. 10-50 has round.80 m long. it to full speed from rest in 6.40 kg is a small ball of mass 1.00 s interval.0 s. During a certain 3. The wheel. 10-48 Problem 74. with a constant angular velocity tal. say. allow himself time to adjust his balance. the Fig. carried 36 wooden cars. give its value. What is the magnitude of his angular acceleration about the start of the 3.0 kg and a rotational inertia of 15. the angular velocity of the cm and rotational inertia 7.0 rad. 10-47. it slows to a stop in 32. it turns has a mass of 70. mass M about any given axis is equal to the rotational inertia of an mation. 10-49 Problem 78.00 mm. from rest at angle u 40° above the horizon- ing. (c) the total energy transferred from mechan. he moves the rotation axis through O? pole to his left (its com moves to the left) to slow the rotation and 70 A wheel. 81 The thin uniform rod in Fig. to his right (his com moves to the What is the rotational inertia of the two-disk assembly about the right) and is in danger of rotating around the wire.0 rev/s. and (d) the number of mass M and radius R about its central axis is equal to the rotational revolutions rotated during the 32. from the beginning of the motion shows two disks that can rotate (let t 0 then). Figure 10-48 gular speed v versus t for the disk. Disk B has been stationary but now begins to rotate at to determine the angular speed of the rod as it Pin a constant angular acceleration of 2.0 rad/s. lines of the two disks have the same Disk A Disk B frictionless pin through one end. from angular cannot be considered to be a point mass for this calculation.00 s interval? (b) How long has the wheel been the wire if his com is 5. Use the principle of conservation of energy θ of 9. What are 78 A rigid body is made of (a) the magnitude of the pulley’s angular acceleration. the reference length 2. the reference lines of the two disks momentarily have the same an. 10-47 Problem 71. He normally carries a long. The string does not slip abruptly ceases.650 rad How many revolutions does it make in this time? in 91.00 rad/s2. Assume that the walker celeration of 2. ** View All Solutions Here ** .5 rad/s. The rod is constrained about a horizontal axis that runs Fig. 72 Attached to each end of a thin steel rod of length 1. each with nitude of either block’s acceleration. about their centers like a merry-go. Ignore air resistance. The body is free to rotate L mass 6. equivalent hoop about that axis.70 s.0 kg m2 about through 90. Fig. At a certain instant. (b) this system is released from rest. The acceleration continues until t 20 s. When stant) angular acceleration in revolutions per minute-squared.0 s. Jr.20 m and 10-49). (c). (c) string tension T1. a wheel undergoes a constant an- string over a pulley of radius 2. (b) Show that the rotational inertia I of any given body of (a). Through what angle does the wheel rotate in the on the pulley. Assuming a constant retarding which the plane of the H is horizontal. the magnitude of the force on the bolt from the axle when the ro. it is rotating at 39. L three identical thin rods. built the original Ferris wheel for the 1893 75 A high-wire walker always attempts to keep his center World’s Columbian Exposition in Chicago. The body is allowed to fall from rest from a position in friction. L string tension T2? gether in the form of a letter H (Fig. to rotate in a horizontal plane about a vertical axis through its mid. 79 SSM (a) Show that the rotational inertia of a solid cylinder of ical energy to thermal energy by friction.600 m. gular displacement u? (b) Will that time t be the first time since t a civil engineering graduate from Rensselaer Problem 81. (a) What is I k . if the hoop has the same mass M 73 A uniform helicopter rotor blade is 7. Disk A is already rotat. AM tor is turning at 320 rev/min? (Hint: For this calculation the blade can be considered to be a point mass at its center of mass.40 gular acceleration. starting from rest. and (d) length L 0. Its not? Assume the mass distribution of a uniform thin rod.0 cm to the right of the wire and (a) he car- turning before the start of the 3.0 s. (e) Now suppose that the re. two 6. When t 2. along the length of one of the legs point. rotates with a constant angular ac.40 103 kg/m3 and a uniform thickness of 5. (The blade 80 A disk rotates at constant angular acceleration.. If any of the quantities axis. (a) What is the angular velocity of the wheel at the wire.40 T1 wheel is 5. compute (a) the angular acceleration. (b). It is released orientation.272 CHAPTER 10 ROTATION ** View All Solutions volume) of 1. heavy ing engineering construction at the time.2 rad/s2.0 ms and the acceleration of the blocks is constant. At time t 0. fastened to. 10-50 82 George Washington Gale Ferris. it is not known interval t 0 to t 40 s? whether there is friction between 77 SSM A record turntable rotating at 3313 rev/min slows down the table and the sliding block. and determine tate without friction about a fixed horizontal axis through its cen. Calculate (a) the angular acceleration of the flywheel. the box has fallen? (b) If the asteroid had. instead. a wheel of radius 0. block that slides on a horizontal frictionless surface. (a) How long does it take the Sun to make one at longitude 20° W? (The resulting tsunamis would have wiped out revolution about the galactic center? (b) How many revolutions has coastal civilization on both sides of the Atlantic. This would have obliterated the city. The Tunguska Event. Fig. tionless horizontal axle. the circle in which the pedals rotate to be 0.40 kg. two circular rings that have a common center and the wheel is attached to a 2. the torque of 960 N m gives the shell an angular acceleration of 6.20 wheel made a complete rotation at constant angular speed in rad/s2 about an axis through the center of the shell.500 tude 12. the other end presses downward on a portion of the can’s top that has been scored. inner radius.20 m P with a speed of 60 m/s and a rotation rate of 90 rad/s.0 kg box. a wheel of radius 0. which is initially at rest. 10-18a.) the Sun completed since it was formed about 4. and (c) the tension T2 in the engine is turned off. When the box has a kinetic energy of 6.0 kg are held together by three rods of negligible mass. at latitude 61° N and longitude 102° (c) the number of revolutions made by the flywheel in stopping. 10-51 round).0 cm. and above remote central Siberia. When you pull upward on one end of the tab. been a metallic asteroid.5 10 9 years ago? 85 A golf ball is launched at an angle of 20° to the horizontal. A massless cord wrapped around the probably the explosion of a stony asteroid about 140 m wide.0900 0. 10-54 Problem 95. The mass. 89 A bicyclist of mass 70 kg puts all his mass on each downward- are connected by a massless cord that is wrapped around a uniform moving pedal as he pedals up a steep road. 1908. Its speed increases at the constant rate of 0. a 30 cm. around a circle 76 m in diameter. force of magnitude P 3. Find (a) the magnitude of the acceleration of the blocks. What is the rotational inertia of the wheel about the axle? 96 Beverage engineering.M. and outer radius of the 94 A car starts from rest and moves around a circular track of rings are given in the following table. and b 50 cm. If a horizontal The construction. of the shell? 83 In Fig. on June 30. The pull Fig. in 20.0 m. what is the magnitude of the angular acceleration of the Fig. (b) 90 The flywheel of an engine is rotating at 25. 10-54 consists of three particles 1 0. determine the number of revolutions the ball makes by tal axis. the cord cannot slip on the disk.0450 connected by massless rods.) ** View All Solutions Here ** . Neglecting is mounted on a frictionless horizon- air drag.050 kg m2. The system is re- Considering only Earth’s rotation. The disk can ro. 10-53. what asteroid would have had to arrive to put the explosion above are (a) the wheel’s rotational kinetic energy and (b) the distance Helsinki at longitude 25° E. E. Problem 86. 10-53 Problem 93. can ro. How much later would such an Milky Way galaxy and is moving in a circle around that center at a asteroid have had to arrive to put the impact in the Atlantic Ocean speed of 250 km/s.0 m/s2.40 kg m2. If M 87 In Fig.20 m is mounted on a fric- 0.300 s.” was about the axis is 0.3 10 4 ly (light-years) from the center of our have reached Earth’s surface.0 rad/s? 2M P 2M ates down the surface at 2. (a) What is the magnitude of its net linear acceleration 15. Take the diameter of disk of mass M 500 g and radius R 12. determine how much later the leased from rest. the flywheel slows at a constant rate and stops cord at the right. The rotational inertia of the the time it reaches maximum height. two blocks. The rotational inertia of the wheel one chance witness “covered an enormous part of the sky. m/s2. Estimate the amount of work that was required of the the rotational inertia of the shell about that axis and (b) the mass machinery to rotate the passengers alone.240 0. When the the tension T1 in the cord at the left. The tab pivots on a central bolt in the can’s top. It is to be 2 0. A tangential force of magni.0 J.120 0. of mass m1 400 g and m2 600 g. approxi- θ mately what is the magnitude of the force applied to the scored Fig. A massless cord is wrapped around the how much work is required to take wheel and attached to a 2. 10-38. 93 SSM A wheel of radius 0.0 kg box that slides on a frictionless sur- the body from rest to an angular a a face inclined at angle u 20° with the horizontal. What are (a) about 2 min. which according to frictionless horizontal axis.0160 0.0 s. wheel? Assume the cord does not slip on the wheel. the magnitude of the maximum torque he exerts about the rota- ter. a huge explosion occurred (b) the angle through which the flywheel rotates in stopping. it could 92 Our Sun is 2.0 N is applied to the block as shown in tate around the common center (like a merry-go. PA R T 1 PROBLEMS 273 ** View All Solutions Here ** each holding up to 60 passengers. tab was a major advance in the engi- neering design of beverage containers.1400 rotated about an axis perpendicular b b to its plane through point P.0 N is applied to the outer edge of the outer ring for 0. The system is released from tion axis of the pedals. 10-52 Problem 87.0 rad/s.0 s What is the change in the angular speed of the construction during later? (b) What angle does this net acceleration vector make with that time interval? the car’s velocity at this time? M Ring Mass (kg) Inner Radius (m) Outer Radius (m) 95 The rigid body shown in Fig.20 m is mounted on a before nuclear weapons.90 m. where another rod of negligible mass lies. 84 At 7⬊14 A. and once all 36 cars were full. If you pull upward with a 10 N force. radius 30. (a) wheel’s circumference is attached to a 6. 10-52. 88 A thin spherical shell has a radius of 1. An applied The cars were loaded 6 at a time. 86 Figure 10-51 shows a flat construction of A massless cord wrapped around Fig. section? (You will need to examine a can with a pull tab. wheel about the axis is 0. The box acceler- speed of 5. the fireball thus created was the brightest flash seen by anyone 91 SSM In Fig.40 m. rest. 780 m long and of negligible mass. The balls may be treated as particles. Pulley A (radius 15 cm) The particles are connected by rods is the drive pulley. C connected by belt 2 to pulley B. the vertical position? ** View All Solutions Here ** . What is 103 In Fig.0 kg. and passes through the center of Fig. The outer radius R of the de. the longer rod? 104 Four particles. linear speed of a point on belt 2. This rigid body tates at 10 rad/s. 10-57. B belts.30 102 N on the ball. di- will the rod momentarily stop in its upward swing? rected opposite its motion.0 m from the end 99 A small ball with mass 1. 102 The rigid object shown in Fig. 10-59 consists of three balls magnitudes a of the centripetal accelera- and three connecting rods. The axis (radius 5 cm) is concentric with Belt 2 body is released from rest with rod pulley B and is rigidly attached AB horizontal (Fig. A can rotate in a vertical plane about dius 10 cm) is connected by Drive a horizontal axis A that passes Rotation belt 1 to pulley A. and it ro. a thin uniform rod (mass 3. 10-61).50 m. 0. Pulley B (ra. Point A is at the C. rods have negligible mass. the linear outer tip of the blade. are placed at the vertices of leys are connected by two Belt 1 a square with sides of length 0. 10-60 Problem 103. each of mass.40 m.60 m. 10-56 Problem 98.30 kg is mounted on one end of a rod of the rod.80 m/s2. to the rod of length 2L. at radial distance Fig. What torque must be applied to the sys- tem to keep it rotating at constant speed? 100 Two thin rods (each of mass d 0. What is the rotational iner. magnitude 0.50 m. The system rotates in a hori- vertical position is 20 J. Fig. (Hint: If the belt between two pulleys does not slip. θ tia of this rigid body about (a) an axis that is perpendicular to the plane of the paper and passes through the center of the shorter _1_ L _1_ L rod and (b) an axis that is perpen.0 m) the rotational inertia of the device about its axis of rotation? rotates freely about a horizontal axis A that is perpendicular to the rod and passes through a point at distance d 1. length 4. and the other has length L2 L2 0. 2 1 2 1 B dicular to the plane of the paper Fig.20 m. 10-56. Determine the rotational kinetic energy 98 A yo-yo-shaped device of the object if it has an angular speed of 1. and the connecting the slope of a plot of a versus radial distance along the blade.20 kg) are joined together to form A a rigid body as shown in Fig. four pul. device which is suspended from a 2M rope wrapped around the hub. (b) the an.50 m. The kinetic energy of the rod as it passes through the 0. Pulley B pulley through one of the particles. has an upward acceleration of Fig.274 CHAPTER 10 ROTATION ** View All Solutions Here ** 97 Figure 10-55 shows a propeller blade A B gular speed of pulley B. 10-58. Pulley C (radius 25 cm) is is the rotational inertia of the body Fig. (c) the angular speed of pulley B. (d) the that rotates at 2000 rev/min about a per.150 m? (b) Find 30.) 1. When a L : constant horizontal force Fapp R θ P M 2L of magnitude 140 N is applied θ to a rope wrapped around the Yo-yo shaped Rope L outside of the device. B' of negligible mass. (a) What A B to it. and lies in the plane of the figure. (a) What is the difference in the Problem 97.20 kg. L 0. (a) about axis A? (b) What is the (linear) speed of the end B of the rod Calculate the rotational inertia of the system about the axis of ro- as the rod passes through the vertical position? (c) At what angle u tation. 10-59 Problem 102. 10-60.50 m. 101 In Fig. 10-55 speeds at the rims of the two pulleys must be equal. 10-57 Problem 100. the box. (b) There is an air drag of 2. is perpendicular kg box as shown in Fig. Fapp Rigid mount that passes through point P and is perpendicular to the plane of the tionless axis is used to lift a 30 figure and (b) an axis that passes through point P. One of the rods has length L1 0.2 rad/s about (a) an axis mounted on a horizontal fric. about axis A? (b) What is the angu- Calculate (a) the linear speed lar speed of the body about axis A when rod AB swings through of a point on belt 1. Hub vice is 0. and (e) the angular speed of pulley pendicular axis at point B. 10-58 Problem 101.6 kg. (a) What is the rotational inertia of the rod zontal circle about the other end of the rod at 5010 rev/min. and tion of point A and of a point at radial distance 0. 10-61 Problem 104. with M 1. and the radius r 2M r of the hub is 0. Fig.