Assertion (A): Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The solid cylinder will reach the bottom of the inclined plane first.
Reason (R): By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.
Solution:
The acceleration of a rolling body down an incline is \( a = \frac{g\sin\theta}{1 + I/(MR^2)} \). For a solid cylinder, \( I = \frac{1}{2}MR^2 \); for a hollow cylinder, \( I = MR^2 \). Since the solid cylinder has a smaller \( I/(MR^2) \) ratio, its acceleration is greater, and it reaches the bottom first.
By conservation of energy, \( Mgh \) converts to kinetic energy, so total KEs are identical if \( M \) and \( h \) are same.
Thus A and R are true, but R does not explain why one reaches first (which depends on the distribution of KE).
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