# Wheel rolling leaned against a vertical wall

I'm new with rigid problems about. I'm trying to solve this:

A massive circular disk of mass m, radius R, and negligible thickness is leaned against a vertical wall, slanting by 45 . In the gravitational field, it can oscillate, rolling without slip on the ground and along the wall.

I need to write down the equations of motion of the disk.

I try to use the Euler's equations. Since there's no torque, $I_1=I_2 \neq I_3$, and $\omega_1=\omega_2=0$ ($\omega_2$ is the desired angular speed) Therefore $$I_2\dot\omega_2=0 \Rightarrow \omega=kte$$

But this is not useful because obviously the motion is an oscillatory one.

How to write the equation of motion of this?

Any suggestions would be very welcome.

I don't want that you solve this for me. I want to understand how to get the equations.

• But there is a torque - what does the normal force do as the disk rocks back and forth? – probably_someone Nov 21 '17 at 19:53
• Oh, ok. I didn't realize that. Can it be written as $Rd\theta mg\cos\phi$? ($\phi$ is the inclination of the disk) – Gabriel Sandoval Nov 21 '17 at 20:01
• It might be easier to just do this from the Lagrangian. The rotational KE in both the $\theta$ and $\phi$ directions is simple to write down, and the gravitational potential depends only on the height of the center of mass, which can be easily converted to a function of $\phi$. – probably_someone Nov 21 '17 at 20:10
• You might want to put a bounty on this question. It is interesting and complicated at once. – Gert Nov 22 '17 at 1:09
• @Gert How i can do it? – Gabriel Sandoval Nov 22 '17 at 3:55