Pendulum-Cart Motion: Solving for Functions of Time

[Perillux]

Homework Statement

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.

Homework Equations

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

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.

 

[anathema]

Dear student,

It seems like you have made some good progress in trying to solve this problem. I would suggest taking a step back and looking at the problem from a different perspective. Instead of trying to relate the motion of the pendulum directly to the motion of the cart, try breaking the problem down into smaller, more manageable parts.

First, consider the motion of the pendulum on its own. Use the equations of motion for a simple pendulum to describe its motion as a function of time. This will give you the angle of the pendulum as it swings back and forth.

Next, think about how the pendulum's motion affects the cart. As the pendulum swings, it will exert a force on the cart due to the tension in the string. This force will cause the cart to accelerate in the direction of the pendulum's motion.

Finally, you can use the equations of motion for a particle to describe the motion of the cart as a function of time. Keep in mind that the acceleration of the cart will be due to both the force exerted by the pendulum and any other external forces acting on the cart.

By breaking the problem down into smaller parts, you can use your knowledge of basic physics principles to solve for the motion of both the pendulum and the cart. Once you have a better understanding of how each component moves, you can then try to extend your solution to the problem involving two pendulums.

I hope this helps and good luck with your problem-solving! Remember, as a scientist, it's important to approach problems with patience, persistence, and a willingness to try different methods.

 

[nlightner]

I would suggest starting by drawing a free body diagram of the system. This will help you visualize the forces acting on the cart and pendulum and how they are related. From there, you can use Newton's laws of motion and the equations you listed to set up a system of equations that describe the motion of both the cart and pendulum as functions of time. You may also want to consider the conservation of energy and momentum in your equations. It may also be helpful to consider the motion of each component separately, and then combine them to find the overall motion of the system. It may take some trial and error, but with persistence and careful calculations, you should be able to find a solution. Good luck!