91Ó°ÊÓ

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 malong the ramp. What was its initial speed?

In Fig. 11-60, a constant horizontal force of magnitude 12 N is applied to a uniform solid cylinder by fishing line wrapped around the cylinder. The mass of the cylinder is 10kg, its radius is 0.10 m, and the cylinder rolls smoothly 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?

In a playground, there is a small merry-go-round of radius 1.20 mand mass 180 kg. Its radius of gyration (see Problem 79 of Chapter 10) is 91.0 cm.A child of mass 44.0 kgruns at a speed of 3.00 m/salong 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 cmand a thickness of 1.2 cm. The density (mass per unit volume) of granite is2.64 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?

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

Wheels Aand Bin Fig. 11-61 are connected by a belt that does not slip. The radius of Bis 3.00 times the radius of A. What would be the ratio of the rotational inertias${\mathbf{I}}_{\mathbf{A}}\mathbf{/}{\mathbf{I}}_{\mathbf{B}}$if the two wheels had (a) the same angular momentum about their central axes? (b) the same rotational kinetic energy?

A uniform wheel of mass 10.0 kgand radius 0.400 mis mounted rigidly on a mass less 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 is0.600 kg.m². The wheel is initially at rest at the top of a surface that is inclined at angle$\mathbf{θ}\mathbf{=}\mathbf{30}\mathbf{.}{\mathbf{0}}^{\mathbf{°}}$. 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 solid sphere of weight 36.0 N rolls up an incline at an angle of$\mathbf{30}\mathbf{.}{\mathbf{0}}^{\mathbf{°}}$. 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?

Question: A hollow sphere of radius 0.15 m, with rotational inertia $\mathit{I}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{040}\mathbf{}{\mathbf{\text{kgm}}}^{\mathbf{\text{2}}}$ about a line through its centre of mass, rolls without slipping up a surface inclined at 30⁰to the horizontal. At a certain initial position, the sphere’s total kinetic energy is 20J. (a) How much of this initial kinetic energy is rotational?

(b) What is the speed of the centre of mass of the sphere at the initial position? When the sphere has moved 1.0 mup the incline from its initial position,

(c) what is its total kinetic energy (d) What are the speed of its centre of mass?

Figure 11-29 shows a particle moving at constant velocity and five points with their xycoordinates. Rank the points according to the magnitude of the angular momentum of the particle measured about them, greatest first.