# Block constrained to move inside a ring

A block of mass m slides on a frictionless table. It is constrained to move inside a ring of radius l which is fixed to the table. at t=0, the block is moving along the inside of the table (i.e., in the tangential direction) with velocity (v)O (v initial) , the coefficient of friction between the block and the ring is "u"

1) find the velocity of the block at later times

2)find the position of the block at later times

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

The normal force exerted by the ring must yield the centripetal acceleration. This means that the normal force, fn, is:

fn = m V^2/l,

directed toward the center of the ring.

where m is the mass of the block.

The frictional force f, is thus:

f = u fn = u m V^2/l,

directed in the opposite direction as the velocity.

Write down Newton's second law:

m dV/dt = - f

This means that:

dV/dt = - u/l V^2 ==>

1/V(t) = (u/l)* t +1/V(0)

I'll leave the rest ( = straightforward integrations and algebra) for you!

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