9.53. Set Up: The hollow cylinder has where and Use coordinates where is upward and at the initial position of the cylinder. Then and where d is the distance it has fallen. Solve: (a) Conservation of energy gives and so and (b) no rotation and (c) In part (a) the cylinder has rotational as well as translational kinetic energy and therefore less translational speed at a given kinetic energy. The kinetic energy comes from a decrease in gravitational potential energy and that is the same, so in (a) the translational speed is less. Reflect: If part (a) were repeated for a solid cylinder, and For a thin-walled hollow cylinder, and Note that all of these answers are independent of the mass m of the cylinder. 9.54. Set Up: If the ball rolls without slipping If there is no friction it slides without rolling and Use coordinates where is upward and at the bottom of the hill, so and Solve: (a) and (b) and 9.55. Set Up: The linear acceleration a of the elevator equals the tangential acceleration of a point on the rim of the
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