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A uniform disc is released from rest at height y1 on a curved trough described by y = k*x². The disc rolls without slipping from A to B, but the trough surface is frictionless from B onward. The disc subsequently rises to a maximum height y2. Find 3*y2 / y1. (Assume the disc radius is very small compared to y1 and y2.)
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Correct answer: 2
Solution
At B, (3/4)mv_B² = mgy1 → v_B² = 4gy1/3. Frictionless region: no torque acts, so omega is constant. At max height y2 translational velocity = 0, rotational KE = (1/2)I*omega² = (1/4)mv_B² is stored. Energy: (1/2)mv_B² + (1/4)mv_B² = mgy2 + (1/4)mv_B² → (1/2)mv_B² = mgy2 → y2 = v_B²/(2g) = (4gy1/3)/(2g) = 2y1/3. So 3y2/y1 = 2.
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