Finding the frequency of a vibrating particle on a string?

[kregg87]

Homework Statement

A particle of mass m is on a massless string of length 3a, which is held horizontally across with a tension T(which you can assume doesn't change with the small vibrations). The particle is a distance of a from one of the edges. Set up a diff. equation that describes the particles motion with time and find its frequency of oscillations.

Homework Equations

mx'' = net force
x''+(ω^2)x=0 => x=Acos(ωt+φ)

The Attempt at a Solution

I originally wrote mx'' = T - mg but this doesn't work since it doesn't involve x and doesn't account for the changing sign of T depending if that particle is above or below the equilibrium point. I tried to describe its position but the best I could do was x(θ) = arctan(x/a) and not x(t) which is what I want (at least I think because then I could take the derivative twice and get the acceleration). Any tips would be appreciated!

 

[Simon Bridge]

Welcome to PF;
Presumably the particle is displaced and then released or something?
You need to start by drawing a diagram of the string and mass when the mass has some arbitrary displacement from it's equilibrium.
Then draw the free body diagram for the mass - notice that T points along the string.
Try to reserve bold-face for vectors only.

 

[kregg87]

Welcome to PF;
Presumably the particle is displaced and then released or something?
You need to start by drawing a diagram of the string and mass when the mass has some arbitrary displacement from it's equilibrium.
Then draw the free body diagram for the mass - notice that T points along the string.
Try to reserve bold-face for vectors only.

That would make more sense... I was only thinking of the component of T parallel to the particles displacement. Thanks!