# Motion Models and Rotating Rod

A particle of mass m is free to slide on a thin rod. The rod rotates in a plane about one end at constant angular velocity w. Show that the motion is given by

r=Ae^(-yt)+Be^(yt),

where y is a constant which you must find and A and B are arbitrary constants. Neglect gravity. Show that for a particular choice of initial conditions, (r(t=0)) and (v(t=0)) it is possible to obtain a solution such that r decreases continually in time, but that for any other choice r will eventually increase. (Exclude cases where the bead hits the origin.)

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#### Solution Preview

The only force present is the reaction of the rod towards the particle that keeps it on the rod is a force that is perpendicular to the rod and hence does not enter the dynamics because the particle can only move along the rod. The acceleration in the radial direction is given by

<br> d^2r/dt^2- r (d theta/dt)^2=0

<br>and is equal to zero, because of no net force along r.

<br>

<br>Now d theta/dt is fixed at the angular velocity w (omega), so we ...