Semi-circle with small angle harmonic oscillations

[Liquidxlax]

Homework Statement

Consider a uniform semicircular disk of radius R, which rolls without slipping on a horizontal surface. Recall that the kinetic energy of an object is the sum of the translational kinetic energy of the centre of mass (point C) and the rotational kinetic energy about the Centre of Mass.

Using Lagrangian methods, show that the angular frequency of small oscillations is

ω = sqrt([8g]/[R(9π -16)])

Homework Equations

.5mR² = I_c + mh² where h = 4R/3π

L = K - U

dL/dq = (d/dt)(dL/dq')

The Attempt at a Solution

First thing to do is find the center of mass of the object

so i solved for that to get (x,y) = (o, 4R/3π)

Then using that I solved for the moment which equals .5mR²

Using the parallel axis thheorem i found that

.5mR² - mh² = I_c

K = .5mv² + .5I_c²

U = mghcos(∅) ( i think)

This has no R dependence so the Euler Lagrange only acts on theta and omega

problem is it's not work, so i obviously messed up on my kinetic and potential energies.

 

[vela]

How are you defining the angle φ? If it's the way I think you're doing it, the potential energy has the wrong sign.

Your expression for the kinetic energy is wrong, though I think it's just a typo. It should be
$$K=\frac{1}{2}mv^2 + \frac{1}{2}I_c \omega^2$$You left out the ω. What is your expression for v in terms of ω?

 

[Liquidxlax]

Yeah i know i didnt transcribe it from my notes well sorry. I did manage to figure it out though. Not sure how to write out phi. Eaaier to draw it