A sphere is released?

Question

A sphere is released from rest at the top of an inclined plane. What is the speed of the sphere at a distance 3.0 m below its starting point. Assume that the sphere rolls without slipping.

Answer 1

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Is the sphere hollow or solid with uniform density? It matters.

If solid, the moment of inertia about its center isI = (2/5)mR²,

and rolling without slipping means that ω = v/R. Then

KE = ½mv² + ½Iω² = ½mv² + ½*(2/5)mR²*(v/R)² = (7/10)mv²

PE becomes KE:

mgh = (7/10)mv²

v = √(10gh/7) = √(10 * 9.8m/s² * 3.0m / 7) = 6.5 m/s ◄

If hollow, then I = (2/3)mR² and KE = (5/6)mv²

leads to v = 5.9 m/s

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