A ring of mass `M` and radius `R` sliding with a velocity `v_(0)` suddenly enters inthrough surface where the coefficient of friction is `mu`, as shown in figure.
.
Choose the correct statement (s).

Options

A. The ring starts its rolling motion when the centre of mass stationary
B. The ring starts rolling motion when the point of contact becomes stationary
C. The time after which the ring starts rolling is `v_(0)//2 mu g`.
D. The rolling velocity is `v_(0)//2`.

Answer

B, C, D

Explanation

Velocity of `c.m` at any time before start rolling `v = v_(0) - mu gt` ….(1)
and for angular velocity
`tau = I alpha = (tau)/(I), omega = alpha t rArr omega = (tau)/(I) t` ….(2)
Let at `t = t_(0)`, ring start pure rolling
from (1),(2) & (3)
`v_(0) - mu g t_(o) = R xx (tau)/(I) t_(0)`
`v - mu g t_(0) = R xx (mu mg R)/(mR^(2)) t_(0)`
`v_(0) - mu g t_(0) = mu g t_(o) rArr t_(o) = (V_(0))/(2 mu g)`.
`(D)` `=v = v_(0) - mu g t_(0) rArr v = v_(0) -(v_(0))/(2)`
`v = (v_(0))/(2)`.