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Glomerular
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Homework Statement
In a uniform gravitational field, there is a uniform solid disk of of mass [itex]M[/itex] and radius [itex]R[/itex]. A point mass [itex]m[/itex] is glued to the disk at a point that is at a distance [itex]a[/itex] from the center of the disk.
The disk rolls without slipping. Find the frequency of small oscillations about the equilibrium point.
I have been told to solve the problem in two ways:
1. Considering two independent bodies, the disk and the point mass.
2. Considering only one body by using their center of mass.
Homework Equations
Euler-Lagrange equations: [itex]\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial {\dot q}}=0[/itex] (1)
Parallel axis theorem : [itex]I=I_{cm}+ MR^2[/itex] (For the second way) (2)
The Attempt at a Solution
First way:
The position of the center of mass of the disk is:
[itex]\begin{cases}
x_{cm}=R\varphi \\
y_{cm}=R
\end{cases}[/itex]
And the position of the point mass is:
[itex]\begin{cases}
x_{p}=x_{cm} - a\sin(\varphi)=R\varphi - a\sin(\varphi)\\
y_{p}=y_{cm} - a\cos(\varphi) =R - a\cos(\varphi)
\end{cases}[/itex]
So I can compute the kinetic energy [itex]K[/itex] and the potential energy [itex]U[/itex] as:
[itex]K=K_{cm}+K_{p}[/itex] and [itex]U=U_{cm}+U_{p}[/itex]
where:
[itex]K_{cm}=\frac{1}{2} M{v^2}_{cm} +\frac{1}{2}I_{disk}\dot{\varphi}^2 = \frac{3}{4}MR^2\dot{\varphi}^2 [/itex]
(I have used [itex]I_{disk}=\frac{1}{2}MR^2[/itex])
[itex]K_{p}=\frac{1}{2} m({\dot{x}^2}_{p}+{\dot{y}^2}_{p} )+\frac{1}{2}I_{p}\dot{\varphi}^2 = \frac{1}{2}mR^2 + \frac{1}{2}ma^2 -maR\cos(\varphi) + \frac{1}{2}ma^2\dot{\varphi}^2[/itex]
(I have used [itex]I_{p}=ma^2[/itex])
[itex]U_{disk}=MgR[/itex]
[itex]U_{p}=mg(R-a\cos(\varphi))[/itex]
Thus, the lagrangian is:
[itex]L=\frac{1}{2}mR^2 + \frac{1}{2}ma^2 -maR\cos(\varphi) + \frac{1}{2}ma^2\dot{\varphi}^2 +\frac{3}{4}MR^2\varphi -MgR - mg(R-a\cos(\varphi)) [/itex]
And using the Euler-Lagrange equation (1) and approximating [itex]sin(\varphi)=\varphi[/itex] I get
[itex]\ddot{\varphi}+{\varphi}[\frac{mRa + mga}{\frac{3}{2}MR^2 +ma^2}]=0[/itex]
Where [itex]\omega^2[/itex] is the term next to [itex]\varphi[/itex] which is not the result I should get.
Where is my error? I have spent hours looking and re-looking at this problem.
Second way:
My problem is that I don't know how to get the rolling condition for this system. The rest should be practically the same as I have done before, but computing the inertia moment by using the parallel axis theorem (2).
Thank you very much, I really appreciate some help on this.
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