Solving for Velocity in a Bead's Circular Motion

[Crater]

Given this set-up.


Consider a bead of mass m that is free to move around a horizontal, circular ring of wire (the wire passes through a hole in the bead). You may neglect gravity in this problem (assume the experiment is being done in space, far away from anything else). The radius of the ring of wire is r. The bead is given an initial speed v_0 and it slides with a coefficient of friction mu_k. In the subsequent steps we will investigate the motion at later times. You should begin by drawing a free-body diagram at some instant of time. Note that there will be a radial acceleration, a_R, and a tangential acceleration, a_T, in this problem.


I'm suppose to find an equation v(t) by combining the equations of radial and tangential accelerations.

With a_R = v^2/r and a_T = dv/dt. I've found a_T to also be equal to N*mu_k with N=m*a_R= mu_k*(v^2/r).

Whenever I try to combine any of these equations and solve for v, I can never get it in terms of other values and t. So I'm at a loss. Any ideas?

 

[mybsaccownt]

well, you know aR and aT

so find the total acceleration and knowing that acceleration is the rate of change of velocity, you can use the relationship

a = dv/dt

to solve for v

by rearranging

a*dt = dv

and

integrating