- #1
Mr Davis 97
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Homework Statement
This problem involves solving a simple differential equation. A block of mass ##m## slides on a frictionless table. It is constrained to move inside a ring of radius ##l## that is fixed to the table. At t = 0, the block is moving along the inside of the ring (in the tangential direction) with velocity ##v_0##. The coefficient of friction between the block and the ring is ##\mu##. Find the velocity of the block at later times
Homework Equations
The Attempt at a Solution
So we start by simply identifying the forces on the body in the plane of rotation (since the normal force on the table cancels with gravitational force). So we have the normal force on the block from the ring, and we have the frictional force between the block and the ring. Using polar coordinates, we find that
In the radial direction:
##-N_r = m(\ddot{r} - r \dot{\theta}^2)##
##\ddot{r} = 0##
Thus
##N_r = mr \dot{\theta}^2##
In the tangential direction:
##-f_{friction} = m(r \ddot{\theta} + 2 \dot{r} \dot{\theta})##
##\dot{r} = 0##
Thus
##f = -mr \ddot{\theta}##
Since we have kinetic friction, ##f = \mu_k N_r##
So
##\mu m r \dot{\theta}^2 = mr \ddot{\theta}##
##\ddot{\theta} + \mu \dot{\theta}^2 = 0##However, this is a nonlinear differential equation, so it can't be the answer. What am I doing wrong?