# Pendulum motion

1. Mar 10, 2009

### Perillux

1. The problem statement, all variables and given/known data
My teacher gave us a problem to work, but it's not assigned. He will probably give extra credit or something if we can do it... But anyway, I kind of want to solve it on my own, but I need a little help getting started.

Ok, so let's say there is a cart of mass 'M' that can move horizontally on frictionless rails, and there is a pendulum of mass 'm' and length 'a' attached to the cart. (either hanging underneath it, or hanging down a rod coming up off the cart, it shouldn't matter.) The pendulum and cart are initially at rest, then suddenly an impulse is applied to the pendulum giving it a momentum of p, and an angular velocity of w. The motion of the pendulum will cause the cart to begin moving.
I need to find a way to describe the motion of the pendulum and cart as functions of time. (I think for the pendulum a function of motion relative to the cart will be fine, or not... either way.)
The actual problem involves two pendulums, but if I can do this I'd like to try and extend it to the two pendulum problem myself.

2. Relevant equations
$\omega = \frac{p}{m*r}$
$a_{centripetal} = \frac{v^{2}}{r}$
F = ma
and possibly:
KE = (1/2)mv^2
PE = mgh

3. The attempt at a solution
I have tried several different approaches, I think that my problem is that I can't figure out exactly how to relate the motion of the pendulum to the resulting motion of the cart.
I tried finding the centripetal force on the pendulum bob, which should be equal in magnitude to the force that will "pull" the cart. There should also be a gravitational force acting parallel to the string (between cart and pendulum). I then add those forces together to find the force on the cart:
$$F_{cart} = -mg*sin^{2}(\theta) + \omega^{2}am*sin(\theta)$$
But theta, is a function of time (I think), so I'm not sure how to proceed.
I also tried a few differential equations that didn't work out so well. I won't list all of my attempts as it will probably just get messy and won't help anyone help me anyway.