A particle of mass m is free to slide on a thin rod. The rod rotates in a plane about one end at a constant angular velocity w. Show that the motion is given by r=Ae^(-γt)+Be^(γt), where γ 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 (that is, 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).
a=(r''-rw^2)êr + (rθ''+2r'we) êθ
The Attempt at a Solution
Ok, for the first part I took twice the derivative of r, and plugged in the radial part of the acceleration (being θ'' and the coriolis force equal to zero, in the stationary frame of reference).
So I only got r''=rw^2, from which I deduced that γ must be w.
From the initial conditions, where t=0, I get
r'(0)=w(B-A)=0 (since I assume it starts with no initial speed), so A=B
From these I get that A=B=r0/2, so the complete solution for the position is
Now, to show that there is a solution where r decreases in time, I figured there should be an initial speed towards the origin, so
r'(0)=w(B-A) does not equals zero, but a certain initial velocity and so
Therefore, the case where r decreases in time should satisfy
Is this correct, or is there another way of showing this ?