A block of mass 'm' sides on a frictionless table. it is constrained to move inside a ring of radius 'R' which is fixed to the table. At t=0, the block is moving along the inside of the ring with velocity 'Vo'. The coefficient of friction between the block and the ring is 'u'.Find the speed and position of the block as the function of 't'.

friction acting on the block at any time is f ...

f = k(normal reaction)                           (k is cofficient of friction)

normal reaction (N)= mv²/R                            (mass is in contact with ring)

f = ma = uN

a = -kv²/R                                     (-ve sign coz ,  friction produces retardation)

now , a  = dv/dt

dv/dt = -k v²/R

dv/v² = -kdt/R    

integrating both sides

(-1/v) = -kt/R + C          ...........1                  ( C is constant of integration)

now at t=0 , v = v_o  (given) so

 C = -1/v_o            ,  after plugging value eq 1 becomes

-1/v = -kt/R - 1/v_o

v = [Rv_o/(R+v_okt)]

Approved

thnx...