A Simple Harmonic Motion Problem

[Faux Carnival]

Homework Statement

[PLAIN]http://img441.imageshack.us/img441/6951/physics.gif

As shown in the figure, one end (O) of a uniform rod of length L and mass M is pivoted to a wall. Two horizontal springs with force constants k₁ and k₂ are attached to the other end (P) of the rod. When the rod is aligned exactly along the vertical axis, the system is in equilibrium.

a) Find the period of small oscillations of the rod.
b) Assume that the initial angle the rod makes with y-axis is a small angle θ and the initial velocity 0. Find the velocity of the end point P of the rod as it passes through the equilibrium position.

Homework Equations

Moment of inertia of a rod relative to its end: I = 1/3 ML²

The Attempt at a Solution

I think I have a solution but it is too long and complex. I doubt it's correct. Any help appreciated.

 

[Swap]

Moment on the rod due to the spring is 2kl^2*theta. Equate it with I(alpha) then u will gte alpha and hence omega. From here u can proceed easily to find the period

 

[Faux Carnival]

That's what I did. But I added the moment from the mass of the rod too.

So k₁L²θ + k₂L²θ + MgLθ/2.

That's equal to Iα. (α being the second derivative of θ)

(k₁L²θ + k₂L²θ + MgLθ/2) / I is therefore w².

That's part a. For part b, w_max = wθ and V_max = wθL.

[PLAIN]http://img375.imageshack.us/img375/6378/captureoj.jpg

Are my answers correct? Thanks.

 

[Swap]

yes u r correct...

 

[Faux Carnival]

Thanks a lot Swap.