Use the worked example above to help you solve this problem. A ball of mass M an

Question

Use the worked example above to help you solve this problem. A ball of mass M and radius R starts from rest at a height of h = 1.59 m and rolls down a theta = 30.5 degree slope, as shown in the figure. What is the linear speed of the ball when it leaves the incline? Assume that the ball rolls without slipping. Repeat this example for a solid cylinder of the same mass and radius as the ball and released from the same height. Mechanical energy is conserved. What is the initial energy of the system? What is the final energy of the system? Write the kinetic energy of the rolling object in terms of angular quantities, but then rewrite it in terms of its mass and speed. Make sure to include a factor (such as the 2/5 in the example) that takes into account the distribution of mass. Your answer will not depend on the mass or the radius of the rolling object. m/s In a race between the two objects on the incline, which one would win? The ball would win. They would tie. The cylinder would win.

Explanation / Answer

I = moment of inertia of cylinder = (0.5) m r2

Using conservation of energy

PE at the Top of incline = KE at the bottom + Rotational KE at bottom

mgh = (0.5) m v2 + (0.5) I w2

mgh = (0.5) m v2 + (0.5) ((0.5) m r2) (v/r)2

mgh = (0.5) m v2 + (0.25) m v2

gh = (0.75) v2

9.8 x 1.59 = (0.75) v2

v = 4.56 m/s

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