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Buffu
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Homework Statement
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 ring with velocity ##v_0##. The coefficient of friction between the bock and ring is ##\mu##, Find the veloctiy of the block at later times.
Homework Equations
The Attempt at a Solution
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Forces due to the ring are ##N^\prime## and ##F_f##
Since in polar coordinates ##\vec a = (\ddot r - r \dot \theta^2)\hat r + \hat\theta(r\ddot\theta + 2\dot r \dot \theta)##
And we know ##r## is constant,
So,
##\vec a = -r\dot \theta^2 \hat r + r\ddot \theta \hat \theta##
Tangential forces are ##m a_\theta## and ##F_f##
So, ##F## and ##F_f##, where ##F## is some forces driving the block forward against friction,
Which gives, ##F - F_f = Mr \ddot \theta = Ml\ddot \theta## ---- (1)
Radial force is ##N^\prime##
##N^\prime = Mr\ddot \theta^2 = Ml\ddot \theta^2##
Since ##N^\prime \mu = F_f##,
Substituting for ##N^prime## in (1)
##F = Ml \ddot \theta + Ml\dot\theta^2 \mu##
What should I do now ? I know I need to solve a DE but I don't how to get that DE.
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