A solid sphere of mass `m` and radius `R` is gently placed on a conveyer belt moving with constant velocity `v_(0)`. If coefficient of friction between belt and sphere is `2//7` the distance traveled by the centre of the sphere before it starts pure rolling is

Options

A. `(v_(0)^(2))/(7g)`
B. `(2v_(0)^(2))/(49g)`
C. `(2v_(0)^(2))/(5g)`
D. `(2v_(0)^(2))/(7g)`

Explanation

`a=(mumg)/(m)=mug=(2)/(7)g`
`alpha=(mumgR)/((2)/(5)mR^(2))`
`=(5mug)/(2R)`
`=(5)/(7)(g)/(R)`
Pure rolling will start when
`v_(P)=v+Romega=v_(0)`
or `at+R(alphat)=v_(0)`
`therefore((2)/(7)gt)+(5)/(7)gt=v_(0)`
`thereforet=(v_(0))/(g)`
`S=(1)/(2)at^(2)`
`=(1)/(2)((2)/(7)g)((v_(0))/(g))^(2)=(v_(0)^(2))/(7g)`