A small hoop of mass `m` is given an initial velocity of magnitude `v_(0)` on the horizontal circular ring of radius `'r'`. If the coefficient of kinetic friction is `mu_(k)` the tangential acceleration of the hoop immediately after is release is `(` assume the horizontal ring to be fixed and not in contact with any supporting surface `)`

Options

A. `mu_(k)g`
B. `mu_(k)(v_(0)^(2))/(r)`
C. `mu_(k)sqrt(g^(2)+(v^(2))/(r))`
D. `mu_(k)sqrt(g^(2)+(v_(0))/(r^(2)))`

Answer

D

Explanation

The free body diagram of hoop is
`:. `The normal reaction `N=sqrt(m^(2)g^(2)+(m^(2)v_(0)^(4))/r^(2))`
`:. ` Frictional force `=mu_(k)N=mu_(k)sqrt(m^(2)g^(2)+(m^(2)v_(0)^(4))/(r^(2)))`

`:. ` tangentail acceleration `=(mu_(k)N)/(m)`
`=mu_(k)sqrt(g^(2)+(v_(0)^(4))/(r^(2)))`