Choose the correct option:
A small collar of mass `m` is given an initial velocity of magnitude `v_(0)` on the horizontal circular track fabricated from a slender rod. If the coefficient of kinetic friction is `mu_(K)`, determine the distance traveled before the collar comes to rest. (Recognize that the friction force depends on the net normal force).

Options

A. `(r )/(mu_(k))ln[(v_(0)^(2)+sqrt(v_(0)^(4)+r^(2)g^(2)))/(rg)]`
B. `(r )/(4mu_(k))ln[(v_(0)^(2)+sqrt(v_(0)^(4)+r^(2)g^(2)))/(rg)]`
C. `(2r)/(mu_(k))ln[(v_(0)^(2)+sqrt(v_(0)^(4)+r^(2)g^(2)))/(rg)]`
D. `(r)/(2mu_(k))ln[(v_(0)^(2)+sqrt(v_(0)^(4)+r^(2)g^(2)))/(rg)]`

Explanation

`-muN=ma`
`-musqrt((mv^(2))/(r )+(mg^(2)))=ma`
`rArr a=-musqrt((v^(4))/(r^(2))+g^(2))`
`rArr (dv)/(dx)(dx)/(dt)=-musqrt((v^(4))/(r^(2))+g^(2))`
`rArr `Let `(v^(2))/(r )=trArr (2vdv)/(r )=dt`
`rArr int((r )/(2)dt)/(sqrt(t^(2)+g^(2)))=-mu intdx`
`rArr (r )/(2){ln(t+sqrt(t^(2)+g^(2)))}=-[mux]_(0)^(s)`
`rArr (r )/(2) {ln(v^(2)/(r )+sqrt((v^(2))/(r^(2))+g^(2)))}_(v_(0))^(0)=-mu[x]_(0)^(s)`
`rArr S=(r )/(2mu)ln[(v_(0)^(2)+sqrt(v_(0)^(4)+r^(2)g^(2)))/(rg)]`