# Lagrangian with springs and rotating wheels

1. Apr 29, 2007

### mekrob

1. The problem statement, all variables and given/known data
A cart of mass M rides on four frictionlesly mounted wheels of radius a and mass m'. The top of the cart is sloped at an angle alpha horizontally and a mass m is suspended from the top of the slope of force constant k. m slides without friction up or down the slope.

a) Write the Lagrangian.
b) What is the frequency of oscillation of the mass m'.

2. Relevant equations
L= T-U
$$I(for disk/wheel)=1/2MR^2$$

3. The attempt at a solution
$$L= 1/2 (4m'+M+m)\dot{x}_{1}^2 + 2I (\omega)^2 + 1/2m(\dot{x_{1}}+\dot{x_{2}} + 2\dot{x_{1}} \dot{x_{2}}cos(\alpha)) + mgx_{2}sin(\alpha) - 1/2kx_{2}^2$$

x1 = position of M
x2 = position of m

Last edited: Apr 30, 2007
2. Apr 29, 2007

### mekrob

3. Apr 29, 2007

### StatusX

What is your question? If you want someone to verify that equation for you, it'd help if you explained how you got it.

4. Apr 29, 2007

### mekrob

This was on an exam (studying for the final). I wrote the same answer on the exam and the only mark the prof made was asking which are $$x_{1}$$ and$$x_{2}$$, which I had forgot to clarify. I received most of the points for the question, so I assume it's right.

5. Apr 29, 2007

### StatusX

We don't do problems for you. Show what you've tried and we'll try to push you in the right direction. Start by writing the equations of motions, and try to figure out how you would extract a frequency from them.

6. Apr 30, 2007

### mekrob

Going through the Lagrangians, I got
$$\ddot {x_{1}} = \frac {gsin(\alpha) - \frac {kx_{2}}{m} - \ddot{x{_2}}}{cos(\alpha)}$$

and

$$(4m' + m + M) \ddot{x_{1}} + 2I\omega^2 + m(\ddot{x_{1}} + \ddot{x_{2}}cos\alpha) = constant$$

Again, I got these right on the exam, apparantely, I was just never good at converting this to $\omega$. Any help getting started?

Last edited: Apr 30, 2007
7. Apr 30, 2007

### StatusX

You need to write $\omega$ (ie, the omega for the rotating wheels, not the one you're trying to find for the oscillation of m) in terms of $x_1$, which will change your second equation. Then this will tell you that $\ddot{x_1}$ is proportional to $\ddot{x_2}$, so you can substitute for $\ddot{x_1}$ in the first equation to get an ODE for $x_2$ which should be easy to solve.