Rolling all the way down

A hollow sphere has a light rope wound over it. The end of the rope is attached to a rigid support which is at a height \(h\) from the surface of the earth, It acquires the same velocity just before touching the ground as that acquired by a free falling body under gravity from a height \(h'\) from the surface of the earth.If \( R_{e} \) is the radius of the earth, what is the value of \( R_{e} [\dfrac{3}{h'} - \frac{5}{h} ] \)?

Details and Assumptions:

  • Radius of sphere is very much smaller than radius of earth

  • Consider the variation of acceleration due to gravity with height as both \(h\) and \(h'\) are quite large

  • The rope doesnt slip on the sphere.

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