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## Homework Statement

A long plank of mass M rests upon a smooth horizontal surface. A thin circular ring (m, R) slips (without rotation) upon the plank with initial velocity v(i). The coefficient of friction between the wheel and the plank is C. at time t, the ring stops slipping and pure rolling starts, find value of t and velocity of center of wheel at t.

## Homework Equations

taking v1 as final velocity of ring, v2 as final velocity of plank, a1 as acceleration of ring and a2 as acceleration of plank, (alpha) as angular acceleration of ring, w as final angular velocity of ring and friction=Cmg. I wrote the following equations-

1) mv1+Mv2=mv(i) (conservation of momentum in horizontal direction)

2) from torque=M.I*(alpha), alpha=Cg/R

3) from translational motion of plank Cmg=Ma2

4) from translational motion of ring Cg=a1

5) w=(Cg/R)*t

## The Attempt at a Solution

I can't figure out the direction of friction and therefore of final angular velocity to use in the no slip condition

v1+Rw=v2, or the directions of a1 and a2 to use in the equations of motion v1=v(i)+a1t. Intuitively, for ring to start rolling, w has to be in a clockwise direction but if plank is moving forward then it can also be in the anticlockwise direction, please shed some light on this and let me know if any of the equations I wrote are incorrect