Calculating Frequency of Oscillation of Plank on Rotating Wheels

[noblesavage8]

I have encountered a question regarding a plank shifting in simple harmonic motion on top of two rotating wheels, rotating in exact opposite directions with the same angular velocities and the question requires me to determine the frequency of oscillation, which has got me stuck. I proved that the plank first of all does go into SHM by using the concept of torque and such but I can't seem to calculate its frequency...
So far, I've come up with:
a[x(t)] = -ug2x(t), where a(u) is the fnction of acceleration with respect to the position of the plank at a given moment, u is the coefficient of kinetic friction between the plank and the wheel, and g is gravity.
And also, I've been fiddling with the equation a(t) = -w^2x(t) and w=2pif and the other SHM equations but I seem to just have too many unknowns and too few equations.
If somebody could just point me in the right direction, that would be greatly appreciated

 

[James R]

Since

$$a = -\omega^2 x$$

and you seem to have

$$a = -ug2x$$

then you can get $\omega$ in terms of your other parameters. Then, the frequency is just $\omega / 2\pi$.

 

[noblesavage8]

see, i arrived at that answer but it seemed strange to me because all it seemed to prove was that f = f, since w = 2pif, so w/2pi is just..well, f. i had expected to find the frequency in terms of the displacement of the block somehow...