Solved: Nonuniform cylindrical object. In Fig. 11-39, a

A car travels at 80 kmlh on a level road in the positive direction of an x axis. Each tire has a diameter of 66 cm. Relative to a woman riding in the car and in unit-vector notation, what are the velocity 11 at the (a) center, (b) top, and (c) bottom of the tire and the magnitude a of the acceleration at the (d) center, (e) top, and (f) bottom of each tire? Relative to a hitchhiker sitting next to the road and in unit- vector notation, what are the velocity 11 at the (g) center, (h) top, and (i) bottom of the tire and the magnitude a of the acceleration at the (j) center, (k) top, and (1) bottom of each tire?

An automobile traveling at 80.0 km/h has tires of 75.0 cm diameter. (a) What is the angular speed of the tires about their axles? (b) If the car is brought to a stop uniformly in 30.0 complete turns of the tires without skidding), what is the magnitude of the angular acceleration of the wheels? (c) How far does the car move during the braking?

A 140 kg hoop rolls along a horizontal floor so that the hoop's center of mass has a speed of 0.150 mls. How much work must be done on the hoop to stop it?

A uniform solid sphere rolls down an incline. (a) What must be the incline angle if the linear acceleration of the center of the sphere is to have a magnitude of 0.10g? (b) If a frictionless block were to slide down the incline at that angle, would its acceleration magnitude be more than, less than, or equal to 0.10g? Why?

H.W A 1000 kg car has four 10 kg wheels. When the car is moving, what fraction of its total kinetic energy is due to rotation of the wheels about their axles? Assume that the wheels have the same rotational inertia as uniform disks of the same mass and size. Why do you not need to know the radius of the wheels?

Figure 11-30 gives the speed v versus time t for a 0.500 kg object of radius 6.00 cm that rolls smoothly down ';6 a 30 ramp. The scale on the ve- E. '" locity axis is set by Vs = 4.0 mls. '" What is the rotational inertia of the object?

In Fig. 11-31, a solid cylinder of radius 10 cm and mass 12 kg starts from rest and rolls without slipping a distance L = 6.0 m down a roof that is inclined at the angle 8 = 30. (a) What is the angular speed of the cylinder about its center as it leaves the roof? (b) The roof's edge is at height H = 5.0 m. How far horizontally from the roof's edge does the cylinder hit the level ground?

Figure 11-32 shows the potential energy U(x) of a solid ball that can roll along an x axis. The scale on the U axis is set by Us = 100 1. The ball is uniform, rolls smoothly, and has a mass of 0.400 kg. It is released at x = 7.0 m headed in the negative direction of the x axis with a mechanical energy of 75 1. (a) If the ball can reach x = 0 m, what is its speed there, and if it cannot, what is its turning point? Suppose, instead, it is headed in the positive direction of the x axis when it is released at x = 7.0 m with 75 1. (b) U(J) U s ... I I If the ball can reach x = 13 m, what is its speed there, and if it cannot, what is its turning point?

rolls smoothly from rest (starting at height H = 6.0 m) until it leaves the horizontal section at the end of the track, at height h = 2.0 m. How far horizontally from point A does the ball hit the floor?

A hollow sphere of radius 0.15 m, with rotational inertia I = 0.040 kg m2 about a line through its center of mass, rolls without slipping up a surface inclined at 30 to the horizontal. At a certain initial position, the sphere's total kinetic energy is 201. (a) How much of this initial kinetic energy is rotational? (b) What is the speed of the center of mass of the sphere at the initial position? When the sphere has moved 1.0 m up the incline from its initial position, what are (c) its total kinetic energy and (d) the speed of its center of mass?

In Fig. 11-34, a constant horizontal force F.pp of magnitude 10 N is applied to a wheel of mass 10 kg and radius 0.30 m. The wheel rolls smoothly on the horizontal surface, and the acceleration of its center of mass has magnitude 0.60 mls2 (a) In unit-vector notation, what is the frictional force on F,.pp Fig. 11 -34 Problem 11. the wheel? (b) What is the rotational inertia of the wheel about the rotation axis through its center of mass?

In Fig. 11-35, a solid brass ball of mass 0.280 g will roll smoothly along a loop-the-Ioop track when released from rest along the straight section. The circular loop has radius R = 14.0 cm, and the ball has radius r

Nonuniform ball. In Fig. 11-36, a ball of mass M and radius R rolls smoothly from rest down a ramp and onto a circular loop of radius 0.48 m. The initial height of the ball is h = 0.36 m. At the loop Fig. 11 -36 Problem 13. bottom, the magnitude of the normal force on the ball is 2.00Mg. The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form 1= {3MR2, but {3 is not 0.4 as it is for a ball of uniform density. Determine {3.

P so that it rolls smoothly along a horizontal path, up along a ramp, and onto a plateau. Then it leaves the plateau horizontally to land on a game board, at a horizontal distance d from the right edge of the plateau. The vertical heights are hI = 5.00 cm and h2 = 1.60 cm. With what speed must the ball be shot at point P for it to land at d = 6.00 cm? Fig. 11-37 Problem 14.

A bowler throws a bowling ball of radius R = 11 cm along a lane. The ball (Fig. 11-38) slides on the lane with initial speed vcom,o = 8.5 mls and initial angular speed Wo = O. The coefficient of kinetic friction between the ball and the lane is 0.21. The kinetic frictional force 7k acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular ac- x celeration of the ball. When speed Fig. 11 -38 Problem IS. Vcom has decreased enough and angular speed w has increased enough, the ball stops sliding and then smoothly. (a) What then is Vcom in terms of w? During the sliding, what are the baH's (b) linear acceleration and (c) angular acceleration? (d) How long does the ball slide? (e) How far does the ball slide? (f) What is the linear speed of the ball when smooth rolling begins?

Nonuniform cylindrical object. In Fig. 11-39, a cylindrical object of mass M and radius R rolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance d = 0.506 m from the end of the ramp. The initial height of the object is H = 0.90 m; the end of the ramp is at height h = 0.10 m. The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form 1= {3MR2, but {3 is not 0.5 as it is for a cylinder of uniform density. Determine {3.

A yo-yo has a rotational inertia of 9S0 g' cm2 and a mass of 120 g. Its axle radius is 3.2 mm, and its string is 120 cm long. The yo-yo rolls from rest down to the end of the string. (a) What is the magnitude of its linear acceleration? (b) How long does it take to reach the end of the string? As it reaches the end of the string, what are its ( c) linear speed, (d) translational kinetic energy, (e) rotational kinetic energy, and (f) angular speed?

In 1980, over San Francisco Bay, a large yo-yo was released from a crane. The 116 kg yo-yo consisted of two uniform disks of radius 32 cm connected by an axle of radius 3.2 cm. What was the magnitude of the acceleration of the yo-yo during (a) its fall and (b) its rise? (c) What was the tension in the cord on which it rolled? (d) Was that tension near the cord's limit of S2 kN? Suppose you build a scaled-up version of the yo-yo (same shape and materials but larger). (e) Will the magnitude of your yo-yo's acceleration as it falls be greater than, less than, or the same as that of the San Francisco yo- yo? (f) How about the tension in the cord?

In unit-vector notation, what is the net torque about the origin on a flea located at coordinates (0, -4.0 m, S.O m) when forces Pi = (3.0 N)k and P2 = (-2.0 N)} act on the flea?

A plum is located at coordinates (-2.0 m, 0, 4.0 m). In unit vector notation, what is the torque about the origin on the plum if that torque is due to a force P whose only component is (a) Ft = 6.0 N,(b) Fx = -6.0 N, (c) Fz = 6.0 N,and (d) Fz = -6.0 N?

In unit-vector notation, what is the torque about the origin on a particle located at coordinates (0, -4.0 m, 3.0 m) if that torque is due to (a) force Pi with components Fix = 2.0 N, Fly = Flz = 0, and (b) force P2 with components F2t = 0, F2y = 2.0 N, F2z = 4.0 N?

A particle moves through an xyz coordinate system while a force acts on the particle. When the particle has the position vector r = (2.00 m)i - (3.00 m)] + (2.00 m)k, the force is given by P = F) + (7.00 N)} - (6.00 N)k and the corresponding torque about the origin is T = (4.00 N . m)i +(2.00 N . m)} - (1.00 N . m)k. Determine Ft.

Force P = (2.0 N)i - (3.0 N)k acts on a pebble with position vector r = (0.50 m)} - (2.0 m)k relative to the origin. In unitvector notation, what is the resulting torque on the pebble about (a) the origin and (b) the point (2.0 m, 0, -3.0 m)?

In unit-vector notation, what is the torque about the origin on a jar ofjalapefio peppers located at coordinates (3.0 m, -2.0 m, 4.0m) due to (a) force Pi = (3.0N)i - (4.0N)] + (S.ON)k, (b) force P2 = (-3.0 N)i - (4.0 N)] - (S.O N)k, and (c) the vector sum of Pi and P2? (d) Repeat part (c) for the torque about the point with coordinates (3.0 m, 2.0 m, 4.0 m).

Force P = (-8.0 N)i + (6.0 N)] acts on a particle with position vector r = (3.0 m)i + (4.0 m). What are (a) the torque on the particle about the origin, in unit-vector notation, and (b) the angle between the directions of rand P?

At the instant of Fig. 11-40, a 2.0 kg particle P has a position vector r of magnitude 3.0 m and angle 81 = 4So and a velocity vector v of magnitude 4.0 mls and angle 82 = 30. Force F, of magnitude 2.0 Nand angle 83 = 30, acts on P. All three vectors lie in the xy plane. About the origin, what are the (a) magnitude and (b) direction of the angular momentum of P and the (c) magnitude and (d) direction of the torque acting on P?

At one instant, force P = 4.0} N acts on a 0.25 kg object that has position vector r = (2.oi - 2.0k) m and velocity vector v = (-s.oi + S.Ok) m/s. About the origin and in unit-vector notation, what are (a) the object's angular momentum and (b) the torque acting on the object?

A 2.0 kg particle-like object moves in a plane with velocity components Vx = 30 mls and Vy = 60 mls as it passes through the point with (x, y) coordinates of (3.0, -4.0) m. Just then, in unitvector notation, what is its angular momentum relative to (a) the origin and (b) the point located at (-2.0, -2.0) m?

particles move in an xy plane. Particle PI has mass 6.S kg and speed VI = 2.2 mis, and it is at distance dl = loS m from point O. Particle P2 has mass 3.1 kg and speed V2 = 3.6 mis, and it is at distance d2 = 2.8 m from point O. What are the (a) magnitude and (b) direction of the net angular momentum of the two particles about O?

wo particles about O? At the instant the displacement of a 2.00 kg object relative to the origin is d = (2.00 m)i + (4;00 m)] - (3.QO m)k, its velocity is v = -(6.00 rnIs)i + (3.00 rnIs)j + (3.00 rnIs)k and it is subject to a force F = (6.00 N)i (8.00 N)] +(4.00 N)k. Find (a) the acceleration of the object, (b) the angular momentum of the object about the origin, (c) the torque about the origin acting on the object, and (d) the angle between the velocity of the object and the force acting on the object.

In Fig. 11-42, a 0.400 kg baIl is shot directly upward at initial speed 40.0 m/s. What is its angular momentum about P, 2.00 m horizontaIly from Fig. 11-42 Problem 31. the launch point, when the baIl is (a) at maximum height and (b) halfway back to the ground? What is the torque on the baIl about P due to the gravitational force when the baIl is ( c) at maximum height and (d) halfway back to the ground?

A particle is acted on by two torques about the origin: 71 has a magnitude of 2.0 N . m and is directed in the positive direction of the x axis, and 72 has a magnitude of 4.0 N . m and is directed in the negative direction of the y axis. In unit-vector notation, find dCldt, where C is the angular momentum of the particle about the origin

At time t = 0, a 3.0 kg particle with velocity v = (5.0 rnIs)i - (6.0 rnIs)] is atx = 3.0 m,y = 8.0 m. It is pulled by a 7.0 N force ill the negative x direction. About the origin, what are (a) the particle's angular momentum, (b) the torque actillg on the particle, and ( c) the rate at which the angular momentum is changing?

A particle is to move in an xy plane, clockwise around the origin as seen from the positive side of the z axis. In unit-vector notation, what torque acts on the particle if the magnitude of its angular momentum about the origin is (a) 4.0 kg m2/s, (b) 4.0t2 kg m2/s, (c) 4.0 Vi. kg m2/s, and (d) 4.01(2 kg . m2/s?

At time (, the vector 7 = 4.0t2; (2.0t + 6.0(2)J gives the position of a 3.0 kg particle relative to the origin of an xy coordinate system (7 is in meters and t is in seconds). (a) Find an expression for the torque acting on the particle relative to the origin. (b) Is the magnitude of the particle's angular momentum relative to the origin increasing, decreasing, or unchanging?

Figure 11-43 shows three rotating, uniform disks that are coupled by belts. One belt runs around the rims of disks A and C. Another belt runs around a central hub on disk A and the rim of disk B. The belts move smoothly without slippage on the rims and hub. Disk A has radius R; its hub has radius 0.5000R; disk B has radius 0.2500R; and disk C has radius 2.000R. Disks Band C have the (C$t2CCJ A C Fig. 11 -43 Problem 36.same density (mass per unit volume) and thickness. What is the ratio of the magnitUde of the angular momentum of disk C to that of diskB?

of mass m = 23 g are fastened to three rods of length d = 12 cm and negligible mass. The rigid assembly rotates around point 0 at the angular speed (J) = 0.85 rad/s. 1ll About 0, what are (a) the rota- Fig. 11-44 Problem 37. tional inertia of the assembly, (b) the magnitUde of the angular momentum of the middle particle, and (c) the magnitude of the angular momentum of the asssembly?

A sanding disk with rotational inertia 1.2 X 10-3 kg m2 is attached to an electric driII whose motor delivers a torque of magnitude 16 N . m about the central axis of the disk. About that axis and with the torque applied for 33 ms, what is the magnitude of the (a) angular momentum and (b) angular velocity of the disk?

SSM The angular momentum of a flywheel having a rotational inertia of 0.140 kg m2 about its central axis decreases from 3.00 to 0.800 kg m2/s in 1.50 s. (a) What is the magnitude of the average torque acting on the flywheel about its central axis during this period? (b) Assuming a constant angular acceleration, through what angle does the flywheel turn? (c) How much work is done on the wheel? (d) What is the average power of the flywheel?

A disk with a rotational inertia of 7.00 kg m2 rotates like a merry-go-round while undergoing a variable torque given by T = (5.00 + 2.00t) N m. At time ( = 1.00 s, its angular momentum is 5.00 kg m2 /s. What is its angular momentum at t = 3.00 s?

ture consistillg of a circular hoop of radius R and mass m, and a square made of four thin bars, each of length R and mass m. The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of 2.5 s. Assuming R = 0.50 m and m = 2.0 kg, calculate (a) the structure's rotational inertia about the axis of rotation and (b) its angular momentum about that axis.

Figure 11-46 gives the torque T that acts on an initially stationary disk that can rotate about its center like a merry-go-round. The scale on the 7 axis is set by 7s = 4.0 N . m. What is the angular momentum of the disk about the rotation axis at times (a) t = 7.0 s and (b) t = 20 s?

In Fig. 11-47, two skaters, each of mass 50 kg, approach each other along parallel paths separated by 3.0 m. They have opposite velocities of 1.4 mls each. One skater carries one end of a long pole of negligible mass, and the other skater grabs the other Fig. 11-47 Problem 43. end as she passes. The skaters then rotate around the center of the pole. Assume that the friction between skates and ice is negligible. What are (a) the radius of the circle, (b) the angular speed of the skaters, and (c) the kinetic energy of the two-skater system? Next, the skaters pull along the pole until they are separated by 1.0 m. What then are (d) their angular speed and (e) the kinetic energy of the system? (f) What provided the energy for the increased kinetic energy?

A Texas cockroach of mass 0.17 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius 15 cm, rotational inertia 5.0 X 10-3 kg m2, and frictionless bearings. The cockroach's speed (relative to the ground) is 2.0 mis, and the lazy Susan turns clockwise with angular speed Wo = 2.8 rad/s. The cockroach finds a bread crumb on the rim and, of course, stops. (a) What is the angular speed of the lazy Susan after the cockroach stops? (b) Is mechanical energy conserved as it stops?

A man stands on a platform that is rotating (without friction) with an angular speed of 1.2 rev/s; his arms are outstretched and he holds a brick in each hand. The rotational inertia of the system consisting of the man, bricks, and platform about the central vertical axis of the platform is 6.0 kg m2 If by moving the bricks the man decreases the rotational inertia of the system to 2.0 kg m2, what are (a) the resulting angular speed of the platform and (b) the ratio of the new kinetic energy of the system to the original kinetic energy? (c) What source provided the added kinetic energy?

The rotational inertia of a collapsing spinning star drops to ~ its initial value. What is the ratio of the new rotational kinetic energy to the initial rotational kinetic energy?

A track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis (Fig. 11-48). A toy train of mass 111 is placed on the track and, with Fig. 11 -48 Problem 47. the system initially at rest, the train's electrical power is turned on. The train reaches speed 0.15 mls with respect to the track. What is the angular speed of the wheel if its mass is 1.1111 and its radius is 0.43 m? (Treat the wheel as a hoop, and neglect the mass of the spokes and hub.)

A Texas cockroach first rides at the center of a circular disk that rotates freely like a merry-goround without external torques. o Radial distance The cockroach then walks out to the edge of the disk, at radius R. Figure 11-49 gives the angular speed W of the cockroach-disk system during the walk. The scale on the W axis is set by Wa = 5.0 rad/s and Wb = 6.0 rad/s. When the cockroach is on the edge at radius R, what is the ratio of the bug's rotational inertia to that of the disk, both calculated about the rotation axis?

Tho disks are mounted (like a merry-go-round) on low-friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with rotational inertia 3.30 kg m2 about its central axis, is set spinning counterclockwise at 450 rev/min. The second disk, with rotational inertia 6.60 kg m2 about its central axis, is set spinning counterclockwise at 900 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 900 rev/min, what are their (b) angular speed and (c) direction of rotation after they couple together?

The rotor of an electric motor has rotational inertia Im = 2.0 X 10-3 kg m2 about its central axis. The motor is used to change the orientation of the space probe in which it is mounted. The motor axis is mounted along the central axis of the probe; the probe has rotational inertia Ip = 12 kg . m2 about this axis. Calculate the number of revolutions of the rotor required to turn the probe through 30 about its central axis.

A wheel is rotating freely at angular speed 800 rev/min on a shaft whose rotational inertia is negligible. A second wheel, initially at rest and with twice the rotational inertia of the first, is suddenly coupled to the same shaft. (a) What is the angular speed of the resultant combination of the shaft and two wheels? (b) What fraction of the original rotational kinetic energy is lost?

A cockroach of mass /11 lies on the rim of a uniform disk of mass 4.00/11 that can rotate freely about its center like a merrygo-round. Initially the cockroach and disk rotate together with an angular velocity of 0.260 rad/s. Then the cockroach walks halfway to the center of the disk. (a) What then is the angular velocity of the cockroach-disk system? (b) What is the ratio K/Ko of the new kinetic energy of the system to its initial kinetic energy? (c) What accounts for the change in the kinetic energy?

A uniform thin rod of length 0.500 m and mass 4.00 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.00 g bullet Axis traveling in the rotation plane is fired Fig. 11 -50 Problem 53. into one end of the rod. As viewed from above, the bullet's path makes angle () = 60.0 with the rod (Fig. 11-50). If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the bullet's speed just before impact?

Figure 11-51 shows an overhead view of a ring that can ro- \ tate about its center like a merry- \ go- round. Its outer radius R2 is 0.800 m, its inner radius R1 is R2/2.00, its mass M is 8.00 kg, and the mass of the crossbars at its center is negligible. It initially rotates at an angular speed of 8.00 rad/s with a cat of Fig. 11-51 Problem 54. mass m = M/4.00 on its outer edge, at radius R z. By how much does the cat increase the kinetic energy of the cat-ring system if the cat crawls to the inner edge, at radius RJ?

A horizontal vinyl record of mass 0.10 kg and radius 0.10 m rotates freely about a vertical axis through its center with an angular speed of 4.7 rad/s. The rotational inertia of the record about its axis of rotation is 5.0 X 10-4 kg . mZ. A wad of wet putty of mass 0.020 kg drops vertically onto the record from above and sticks to the edge of the record. What is the angular speed of the record immediately after the putty sticks to it?

In a long jump, an athlete leaves the ground with an initial angular momentum that tends to rotate her body forward, threatening to ruin her landing. To counter this tendency, she rotates her outstretched arms to "take up" the angular momentum (Fig. 11-18). In 0.700 s, one arm sweeps through 0.500 rev and the other arm sweeps through 1.000 rev. Treat each arm as a thin rod of mass 4.0 kg and length 0.60 m, rotating around one end. In the athlete's reference frame, what is the magnitude of the total angular momentum of the arms around the common rotation axis through the shoulders?

A uniform disk of mass 10m and radius 3.0r can rotate freely about its fixed center like a merry-go- round. A smaller uniform disk of mass m and radius r lies on top of the larger disk, concentric with it. Initially the two disks rotate together with an angular velocity of 20 rad/s. Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their angular velocity about the center of the larger disk? (b) What is the ratio K/Ko of the new kinetic energy of the two-disk system to the system's initial kinetic energy?

A horizontal platform in the shape of a circular disk rotates on a frictionless bearing about a vertical axle through the center of the disk. The platform has a mass of 150 kg, a radius of 2.0 m, and a rotational inertia of 300 kg . mZ about the axis of rotation. A 60 kg student walks slowly from the rim of the platform toward the center. If the angular speed of the system is 1.5 rad/s when the student starts at the rim, what is the angular speed when she is 0.50 m from the center?

Figure 11-52 is an overhead view of a thin uniform rod of length 0.800 m and mass M rotating horizontally at angular speed 20.0 rad/s about . Rota~onJ axIS an axis through its center. A particle Fig. 11 -52 Problem 59. of mass M/3.00 initially attached to t ! one end is ejected from the rod and travels along a path that is perpendicular to the rod at the instant of ejection. If the particle's speed vp is 6.00 mls greater than the speed of the rod end just after ejection, what is the value of v p ?

In Fig. 11-53, a 1.0 g bullet is fired into a 0.50 kg block attached to the end of a 0.60 m nonuniform rod of mass 0.50 kg. The block-rod-bullet system then rotates in the plane of the figure, about a fixed axis at A. The rotational inertia of the rod alone about that axis at A is 0.060 kg mZ. Treat the Bullet Block block as a particle. (a) What then is the rotational inertia of the block-rod-bullet system about point A? (b) If the angular speed of the system about A just after impact is 4.5 rad/s, what is the bullet's speed just before impact?

The uniform rod (length 0.60 m, mass 1.0 kg) in Fig. 11-54 rotates in the plane of the figure about an axis through one end, with a rotational inertia of 0.12 kg mZ. As the rod swings through its lowest position, it collides with a 0.20 kg putty wad that sticks to the end of the rod. If the rod's angular speed just before ---4. : collision is 2.4 rad/s, what is the angu- . I lar speed of the rod-putty system Rotation axis Rod immediately after collision?

During a jump to his partner, an aerialist is to make a quadruple somersault lasting a time t = 1.87 s. For the first and last quarter-revolution, he is in the extended orientation shown in Fig. 11-55, with rotational inertia 11 = 19.9 kgm2 around his center of mass (the dot). During the rest of the flight he is in a tight tuck, with rotational inertia 1z = 3.93 kg mZ. What must be his angular speed Wz around his center of mass during the tuck?

stands on the edge of a stationary merry-go-round of radius 2.0 m. The rotational inertia of the merry- go-round about its rotation axis is 150 kg m2 The child catches a ball of mass 1.0 kg thrown by a friend. Just before the ball is caught, it has a horizontal velocity 11 of mag- I I I I I I~ : , 1>-1 I Child nitude 12 mis, at angle if; = 37 with Fig. 11 -56 Problem 63. a line tangent to the outer edge of the merry-go-round, as shown. What is the angular speed of the merry-go-round just after the ball is caught?

A ballerina begins a tour jete (Fig. 11-19a) with angular speed Wi and a rotational inertia consisting of two parts: ~eg = 1.44 kg mZfor her leg extended outward at angle (J = 90.0 to her body and .4runk = 0.660 kg mZ for the rest of her body (pri- marily her trunk). Near her maximum height she holds both legs at angle e = 30.0 to her body and has angular speed wf (Fig. 11- 19b). Assuming that ftrunk has not changed, what is the ratio Wf/Wi?

attached to the ends of a thin rod of length 50.0 cm and negligible mass. The rod is free to rotate in a vertical plane without friction about a horizontal axis through its ... R ota~on\ axiS (). Putty wad center. With the rod initially horizon- Fig. 11 -57 Problem 65. tal (Fig. 11-57), a 50.0 g wad of wet putty drops onto one of the balls, hitting it with a speed of 3.00 mls and then sticking to it. (a) What is the angular speed of the system just after the putty wad hits? (b) What is the ratio of the kinetic energy of the system after the collision to that of the putty wad just before? (c) Through what angle will the system rotate before it momentarily stops?

In Fig. 11-58, a small 50 g block slides down a frictionless surface through height h = 20 cm and then sticks to a uniform rod of mass 100 g and length 40 cm. The rod pivots about point 0 through angle ebefore momentarily stopping. Find e.

Figure 11-59 is an overhead view of a thin uniform rod of length o 0.600 m and mass M rotating hori- Fig. 11 -58 Problem 66. zontally at 80.0 rad/s counterclockwise about an axis through its center. A particle of mass M/3.00 and traveling horizontally at speed 40.0 m/s hits the rod and sticks. The particle's path is perpendicular to the rod at the instant of the hit, at a distance d from the rod's center. (a) At what value of d are rod and particle stationary after the hit? (b) In which direction do rod and particle rotate if d is greater than this value?

A top spins at 30 rev/s about an axis that makes an angle of 30 with the vertical. The mass of the top is 0.50 kg, its rotational inertia about its central axis is 5.0 X 10-4 kg m2, and its center of mass is 4.0 cm from the pivot point. If the spin is clockwise from an overhead view, what are the (a) precession rate and (b) direction of the precession as viewed from overhead?

A certain gyroscope consists of a uniform disk with a 50 cm radius mounted at the center of an axle that is 11 cm long and of negligible mass. The axle is horizontal and supported at one end. If the disk is spinning around the axle at 1000 rev/min, what is the precession rate?

A uniform solid ball rolls smoothly along a floor, then up a ramp inclined at 15.0. It momentarily stops when it has rolled 1.50 m along the ramp. What was its initial speed?

In Fig. 11-60, a constant -> -> Fapp horizontal force Papp of magnitude 12 N is applied to a uniform solid line cylinder by fishing line wrapped around the cylinder. The mass of the cylinder is 10 kg, its radius is 0.10 m, x and the cylinder rolls smoothly Fig. 11 -60 Problem 71. on the horizontal surface. (a) What is the magnitude of the acceleration of the center of mass of the cylinder? (b) What is the magnitude of the angular acceleration of the cylinder about the center of mass? (c) In unit-vector notation, what is the frictional force acting on the cylinder?

A thin-walled pipe rolls along the floor. What is the ratio of its translational kinetic energy to its rotational kinetic energy about the central axis parallel to its length?

A 3.0 kg toy car moves along an x axis with a velocity given by v = -2.0t3 i mis, with t in seconds. For t > 0, what are (a) the angular momentum L of the car and (b) the torque T on the car, both calculated about the origin? What are (c) L and (d) T about the point (2.0 m, 5.0 m, O)? What are (e) L and (f) T about the point (2.0 m, - 5.0 m, O)?

A wheel rotates clockwise about its central axis with an angular momentum of 600 kg m2 /s. At time t = 0, a torque of magnitude 50 N . m is applied to the wheel to reverse the rotation. At what time t is the angular speed zero?

In a playground, there is a small merry-go-round of radius 1.20 m and mass 180 kg. Its radius of gyration (see Problem 79 of Chapter 10) is 91.0 cm. A child of mass 44.0 kg runs at a speed of 3.00 mls along a path that is tangent to the rim of the initially stationary merry-go-round and then jumps on. Neglect friction between the bearings and the shaft of the merry-go-round. Calculate (a) the rotational inertia of the merry-go-round about its axis of rotation, (b) the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round, and (c) the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round.

A uniform block of granite in the shape of a book has face dimensions of 20 cm and 15 cm and a thickness of 1.2 cm. The density (mass per unit volume) of granite is 2.64 g/cm3. The block rotates around an axis that is perpendicular to its face and halfway between its center and a corner. Its angular momentum about that axis is 0.104 kg m2 /s. What is its rotational kinetic energy about that axis?

Tho particles, each of mass 2.90 X 10-4 kg and speed 5.46 mis, travel in opposite directions along parallel lines separated by 4.20 cm. (a) What is the magnitude L of the angular momentum of the two- particle system around a point midway between the two lines? (b) Does the value of L change if the point about which it is calculated is not midway between the lines? If the direction of travel for one of the particles is reversed, what would be (c) the answer to part (a) and (d) the answer to part (b)?

A wheel of radius 0.250 m, which is moving initially at 43.0 mis, rolls to a stop in 225 m. Calculate the magnitudes of (a) its lin- ear acceleration and (b) its angular acceleration. (c) The wheel's rotational inertia is 0.155 kg m2 about its central axis. Calculate the magnitude of the torque about the central axis due to friction on the wheel.

Wheels A and B in Fig. 11-61 are connected by a belt that does not slip. The radius of B is 3.00 times the radius of A. What would be the ratio of the rotational inertias lA/Is if the two wheels had (a) the same angular Fig. 11-61 Problem 79. momentum about their central axes and (b) the same rotational kinetic energy

A 2.50 kg particle that is moving horizontally over a floor with velocity (-3.00 m/s)] undergoes a completely inelastic collision with a 4.00 kg particle that is moving horizontally over the floor with velocity (4.50 m/s)i. The collision occurs at xy coordinates (-0.500 m, -0.100 m). After the collision and in unit-vector notation, what is the angular momentum of the stuck-together particles with respect to the origin?

A uniform wheel of mass 10.0 kg and radius 0.400 m is mounted rigidly on a massless axle through its center (Fig. 11-62). The radius of the axle is 0.200 m, and the rotational inertia of the wheel-axle combination about its central axis is 0.600 kg m2 The wheel is initially at rest at the top of a surface that is inclined at angle B = 30.0 with the horizontal; the axle rests on the surface while the wheel extends into a groove in the surface without touching the surface. Once released, the axle rolls down along the surface smoothly and without slipping. When the wheel-axle combination has moved down the surface by 2.00 m, what are (a) its rotational kinetic energy and (b) its translational kinetic energy?

A uniform rod rotates in a horizontal plane about a vertical axis through one end. The rod is 6.00 m long, weighs 10.0 N, and rotates at 240 rev/min. Calculate (a) its rotational inertia about the axis of rotation and (b) the magnitude of its angular momentum about that axis

A solid sphere of weight 36.0 N rolls up an incline at an angle of 30.0. At the bottom of the incline the center of mass of the sphere has a translational speed of 4.90 m/s. (a) What is the kinetic energy of the sphere at the bottom of the incline? (b) How far does the sphere travel up along the incline? (c) Does the answer to (b) depend on the sphere's mass?

Suppose that the yo-yo in Problem 17, instead of rolling from rest, is thrown so that its initial speed down the string is 1.3 m/s. (a) How long does the yo-yo take to reach the end of the string? As it reaches the end of the string, what are its (b) total kinetic energy, (c) linear speed, (d) translational kinetic energy, (e) angular speed, and (f) rotational kinetic energy?

A girl of mass M stands on the rim of a frictionless merry-goround of radius R and rotational inertia I that is not moving. She throws a rock of mass m horizontally in a direction that is tangent to the outer edge of the merry-go-round. The speed of the rock, relative to the ground, is v. Afterward, what are (a) the angular speed of the merry-go-round and (b) the linear speed of the girl?

At time t = 0, a 2.0 kg particle has the position vector r = (4.0 m)i - (2.0 m)] relative to the origin. Its velocity is given by It = (-6.0t2 m/s)i for t:2: 0 in seconds. About the origin, what are (a) the particle's angular momentum land (b) the torque T acting on the particle, both in unit-vector notation and for t> O? About the point (-2.0m, -3.0m, 0), what are (c) l and (d) T for t > O?

If Earth's polar ice caps fully melted and the water returned to the oceans, the oceans would be deeper by about 30 m. What effect would this have on Earth's rotation? Make an estimate of the resulting change in the length of the day.

A 1200 kg airplane is flying in a straight line at 80 mis, 1.3 km above the ground. What is the magnitude of its angular momentum with respec

With axle and spokes of negligible mass and a thin rim, a certain bicycle wheel has a radius of 0.350 m and weighs 37.0 N; it can turn on its axle with negligible friction. A man holds the wheel above his head with the axle vertical while he stands on a turntable that is free to rotate without friction; the wheel rotates clockwise, as seen from above, with an angular speed of 57.7rad/s, and the turntable is initially at rest. The rotational inertia of wheel + man + turntable about the common axis of rotation is 2.10 kg . m2 The man's free hand suddenly stops the rotation of the wheel (relative to the turntable). Determine the resulting (a) angular speed and (b) direction of rotation of the system.

For an 84 kg person standing at the equator, what is the magnitude of the angular momentum about Earth's center due to Earth's rotation?

A small solid sphere with radius 0.25 cm and mass 0.56 g rolls without slipping on the inside of a large fixed hemisphere with radius 15 cm and a vertical axis of symmetry. The sphere starts at the top from rest. (a) What is its kinetic energy at the bottom? (b) What fraction of its kinetic energy at the bottom is associated with rotation about an axis through its com? (c) What is the magnitude of the normal force on the hemisphere from the sphere when the sphere reaches the bottom?

An automobile has a total mass of 1700 kg. It accelerates from rest to 40 km/h in 10 s. Assume each wheel is a uniform 32 kg disk. Find, for the end of the 10 s interval, (a) the rotational kinetic energy of each wheel about its axle, (b) the total kinetic energy of each wheel, and (c) the total kinetic energy of the automobile.

A body of radius R and mass m is rolling smoothly with speed v on a horizontal surface. It then rolls up a hill to a maximum height h. (a) If h = 3v2/4g, what is the body's rotational inertia about the rotational axis through its center of mass? (b) What might the body be?