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Finding the ratio between two velocities down a ramp.

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A marble and a cube are placed at the top of a ramp. Starting from rest at the same height, the marble rolls and the cube slides (no kinetic friction) down the ramp. Find the ratio of the center-of-mass speed of the cube to the center-of-mass speed of the marble at the bottom of the ramp.

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<br>The rolling marble has a friction force acting on it at the point of contact, (otherwise it would slide, not roll). We take the point of contact as a reference axis, and apply 'net torque about that axis = (moment of inertia about that axis) x (angular acceleration of the marble).
<br>The net torque about that axis is:
<br>(1) '(M g ...

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