Pendulum involving Spring and Rotating plank

[kspabo]

Homework Statement

[/B]
In the overhead view of the figure, a long uniform rod of mass m = 0.24 kg is free to rotate in a horizontal plane about a vertical axis through its center. A spring with force constant 240 N/m is connected horizontally between one end of the rod and a fixed wall. When the rod is in equilibrium, it is parallel to the wall. What is the period of the small oscillations that result when the rod is rotated slightly and released?

Homework Equations

Torque = rF
Torque = I*a(angular accel)
Period = 2pi sqrt(I/k)
w(angular vel.) = sqrt(k/m)

The Attempt at a Solution

So I started by trying to related the torque caused by the spring (Where L is the length of the rod):

T=r F = (L/2) (-kx)
x (is the extension of the spring) = (L/2) sinθ
T= (L/2) (-k)((L/2) sinθ) = (L^2/4) (-k) sinθ
T= Ia = ((1/12) mL^2) (a)

(L^2/4) (-kx) sinθ = ((1/12) mL^2) (a)
a = 3(k/m) sinθ

 

[TSny]

Hello.

Your work looks good. I think you dropped a negative sign in the last equation. Since the oscillations are small, you can make the usual approximation for sinθ.