The ratio of the accelerations for a solid sphere (mass `m, and radius R`) rolling down an incline of angle `theta` without slipping, and slipping down the incline without rolling is

Options

A. `2:5`
B. `7:5`
C. `5:7`
D. `2:3`

Answer

C

Explanation

When the sphere rolls down the inclined plane without slipping, its acceleration
`a_(1)=(g sin theta)/(1+(1)/(MR^(2)))`
For the sphere `I=(2)/(5)MR^(2)`
`therefore a_(1)=(g sin theta)/(1+(2)/(5)(MR^(2))/(MR^(2)))=(5)/(7)g sin theta` ….(1)
and when it slips without rolling, its acceleration
`a_(2)=g sin theta " "` ....(2)
`therefore (a_(1))/(a_(2))=(5)/(7)(g sin theta)/(g sin theta)=(5)/(7)`