Question

A block of mass m slides on a frictionless table. It is constrained to move inside a ring of radius R. At time `t=0`, the block is moving along the inside of the ring with velocity `v_o`. The coefficient of friction between the block and the ring `mu`?

Find the speed of the block at time `t`

Answer 1

The above figure presents the situation of sliding of a block of mass m on a frictionless table constrained to move inside a ring of radius R as described in the problem.

Let `v` represents the tangential velocity of the block at an instant of time. At that instant the centrifugal reactionary force on the block is `color(blue)((mv^2)/R)`. So the force of retardation due to friction will be `color(red)((mumv^2)/R)`.

So by Newton's law we can write

`m(dv)/(dt)=-(mumv^2)/R`

`=>(dv)/v^2=-mu/Rdt`

Integrating for `t=0 to t=t` when velocity changes from `v_0to V`

`int_(v_0)^V(dv)/v^2=-mu/Rint_0^tdt`

`=>[v^(-2+1)/(-2+1)]_(v_0)^V=-mu/R[t]_0^t`

`=>[-1/v]_(v_0)^V=-mu/R[t]_0^t`

`=>-1/V+1/v_0=-mu/Rt`

`=>1/V=1/v_0+mu/Rt`

`=>1/V=(R+muv_0t)/(v_0R)`

`=>V=(v_0R)/(R+muv_0t)`

So speed of the block at time t

`color(blue)(V=v_0/(1+(muv_0t)/R))`