A uniform ring of radius `R` placed on a rough horizontal surface is given a sharp impuilse as in figure. As a consequence it acquires a linear velocity of `v m//s`. If coefficient of frication between the ring & the horizontal surface the be `mu`. Then answer the follwing questions based on given information. (Use `g = 10 m//s^(2)`)

Based on above information choose the correct option(s)-

Options

A. The time after which the ring will start pure rolling is `2` sec if `v` is `20 m//s` and `mu` is `0.5`
B. The time after which the ring will start pure rolling is `2` sec if `v` is `10 m//s` and `mu` is `0.25`
C. The time after which the ring will start pure rolling is `4` sec if `v` is `20 m//s` and `mu` is `0.5`
D. frication will change its direction if `v = 20 m//s` and `mu = 0.5`

Answer

A, B

Explanation

Suppose velocity of ring.

When it starts pure rolling is `v'`. Angular moementum is constant about point of contact as there is no external torque acting about it. Thus
`m.v.R = m.R^(2)((v')/(R)) + mv'R`
`rArr v' = (v)/(2)m//s`
using `v = u + at`
`v' = v + (-mu xx g)t rArr t = (v)//(2mug)`sec.