A solid sphere rolls without slipping and presses a spring of spring constant `k` as shown in figure. Then, the compression in the spring will be

Options

A. `v sqrt((2M)/(3k))`
B. `vsqrt((2M)/(5k))`
C. `v sqrt((5k)/(7M))`
D. `v sqrt((7M)/(5k))`

Answer

D

Explanation

In case of pure rolling, ratio of rotational to translational kinetic energy is 2/5. Therefore, total kinetic energy is 7/5 times the translational kinetic energy. At maximum compression, whole of energy is elastic potential. Hence,
`(7)/(5)((1)/(2)Mv^(2))=(1)/(2)kx_(max)^(2)`
`:.` The compression of the spring, `x_(max)=vsqrt((7M)/(5k))`.