Physics Homework #10

1. In the figure, two blocks, of mass m1 = 422 g and m2 = 703 g, are connected by a massless cord that is wrapped around a uniform disk of mass M = 523 g and radius R = 13.7 cm. The disk can rotate without friction about a fixed horizontal axis through its center; the cord cannot slip on the disk. The system is released from rest. Find (a) the magnitude of the acceleration of the blocks, (b) the tension T1 in the cord at the left and (c) the tension T2 in the cord at the right.
2. In the figure, a wheel of radius 0.39 m is mounted on a frictionless horizontal axle. A massless cord is wrapped around the wheel and attached to a 3.1 kg box that slides on a frictionless surface inclined at angle θ = 26 o with the horizontal. The box accelerates down the surface at 2.4 m/s2. What is the rotational inertia of the wheel about the axle?
3. The rigid body shown in the figure consists of three particles connected by massless rods. It is to be rotated about an axis perpendicular to its plane through point P. If M = 0.31 kg, a = 32 cm, and b = 55 cm, how much work is required to take the body from rest to an angular speed of 5.6 rad/s?

4. A yo-yo-shaped device mounted on a horizontal frictionless axis is used to lift a 34 kg box as shown in the figure. The outer radius R of the device is 0.34 m, and the radius r of the hub is 0.13 m. When a constant horizontal force of magnitude 170 N is applied to a rope wrapped around the outside of the device, the box, which is suspended from a rope wrapped around the hub, has an upward acceleration of magnitude 0.91 m/s2. What is the rotational inertia of the device about its axis of rotation?
5. A bowler throws a bowling ball of radius R = 11 cm along a lane. The ball (the figure) slides on the lane with initial speed vcom,0 = 4.5 m/s and initial angular speed ω0 = 0. The coefficient of kinetic friction between the ball and the lane is 0.13. The kinetic frictional force k acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed vcom has decreased enough and angular speed φ has increased enough, the ball stops sliding and then rolls smoothly. During the sliding, what are the ball’s (a) linear acceleration and (b) angular acceleration? (c) How long does the ball slide? (d) How far does the ball slide? (e) What is the linear speed of the ball when smooth rolling begins? Note that the clockwise direction is taken as negative.

6. A plum is located at coordinates (-1.29 m, 0, 7.02 m). In unit-vector notation, what is the torque about the origin on the plum if that torque is due to a force whose only component is (a) Fx = 2.01 N, (b) Fx = -2.01 N, (c) Fz = 2.01 N, and (d) Fz = -2.01 N?
7. Force acts on a particle with position vector . What are (a) the magnitude of the torque on the particle about the origin and (b) the angle between the directions of and ?
8. At one instant, force acts on a 0.468 kg object that has position vector and velocity vector . About the origin and in unit-vector notation, what are (a) the object’s angular momentum and (b) the torque acting on the object?
9. Two 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 4.77 kg•m2 about its central axis, is set spinning counterclockwise (which may be taken as the positive direction) at 312 rev/min. The second disk, with rotational inertia 8.25 kg•m2 about its central axis, is set spinning counterclockwise at 749 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 749 rev/min, what are their (b) angular velocity (using the correct sign for direction) and (c) direction of rotation after they couple together?
10. 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 radius of 4.44 m and a rotational inertia of 410 kg•m2 about the axis of rotation. A 55.2 kg student walks slowly from the rim of the platform toward the center. If the angular speed of the system is 1.40 rad/s when the student starts at the rim, what is the angular speed when she is 0.812 m from the center?
4.6987 Units: rad/s