# Homework Help: Derive velocity function for pendulum

1. Dec 14, 2009

### quantum13

1. The problem statement, all variables and given/known data
A regular pendulum is swinging back and forth. Assuming it starts from a horizontal position, find an expression for its tangential velocity.

(I don't know if the question is posed right, I just asked myself it as a challenge)

2. The attempt at a solution
using a freebody diagram, I know that the tangential vector is the tangential component of weight. let Θ be the angle such that the tangential component of weight is mgsinΘ.

F = ma
mgsinΘ = ma
gsinΘ = dv/dt

I know im supposed to do
v = ∫gsinΘ dt
but I have no clue how to do that except using rotation equations,
Θ = α/2 t^2 + ωt = a/2r t^2 + v/r t = gcosΘ/2r t^2 + v/r t

which leaves me with messy algebra that brings me in circles...

can anyone help me set up this problem?

2. Dec 14, 2009

### AEM

What you might want to do is write v in terms of theta. If l is the length of the pendulum, then $v = l \frac{d\theta}{dt}$. Then the next thing you should do is restrict you pendulum to small oscillations (small values of theta) that way you can use the standard $sin(\theta) \approx \theta$ approximation. Note two things: you will end up with the second derivative of theta, and the equation for simple harmonic motion. You can use the solution of that equation to get the velocity you want from the relationship between the linear and angular velocity I wrote earlier.