How Can We Prove the Time Period Formula of a Simple Pendulum?

[Michaelcarson11]

Homework Statement

As part of my coursework, I need to be able to prove the formula for the time period of a simple pendulum. I know what the formula is but I really don't know how to prove it. I would appreciate any help.

Homework Equations

T = 2*PI*Root(l/g)

The Attempt at a Solution

I have not got very far at all with the proof. I worked out that the force back towards the rest position is -mgsinx and we were told that we needed to use the approximation sinx= x, but I cannot get any further

 

[HallsofIvy]

It's not clear to me what the problem is. You say are to "prove the formula for the time period of a simple pendulum." But then you give as a "relevant equation" precisely that formula. Do you mean you want to derive that equation? From what hypotheses? What I would do is set up the formula "F= ma" in terms of the forces on the pendulum, then solve that equation and derive the formula from that. From what you say, you have derived the equation and it should have "sine" in it. The difficulty is that nonlinear equation does not have an exact solution and, in fact, the formula you give for the period is not exactly true. Replacing sin(x) with x (Approximately true for small x. Do you remember from Calculus that sin(x)/x goes to 1 as x goes to 0.) give you a linear equation with linear coefficients. Please show exactly what you have done so far.