Lagrangian with springs and rotating wheels

[mekrob]

Homework Statement

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'.

Homework Equations

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

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

 

[mekrob]

View attachment untitled.bmp

A rough drawing with paint of the problem.

 

[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.

 

[mekrob]

What is your question?

a) Write the Lagrangian.

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} andx_{2}, which I had forgot to clarify. I received most of the points for the question, so I assume it's right.

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

 

[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.

 

[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?

 

[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.