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# Two Disk problem

Consider a system composed of two plane disks, which we will designate as$$D$$ and $$d$$. Their radii are $$R$$ and $$r$$ respectively $$(R>r)$$. The disk $$d$$ is fixed over the disk $$D$$ and a distance $$b$$ from the center, as in the figure. The disk $$D$$ can spin and move freely in a friction-free platform, while the other disk $$d$$ is fixed on a point of disk $$D$$, but can spin freely without friction. No external forces act on the system. The masses of the disks are $$M$$ and $$m$$ respectively.

Demonstrate that the angular velocity of both disks is a constant, in other words: $$\frac{d^{2}\theta}{dt^{2}}=\frac{d^{2}\varphi}{dt^{2}}=0$$. Both angles are defined in the figure.

![Two Disks] (https://dl.dropboxusercontent.com/u/38822094/TwoDisks.JPG)

Attempted solution:

I will start by writing the conservation laws of both disks, in linear momentum by components we have:

$$p_{x}=(M+m){\dot{x}}_{D}\dot{-mb\sin\theta}\dot{\theta}$$

$$p_{y} = (M+m){\dot{y}}_{D}\dot{+mb\cos\theta}\dot{\theta}$$

Then the equation for the conservation of energy:

$$E=\frac{1}{2}M\left(\dot{x}_{D}^{2}+\dot{y}_{D}^{2}\right)+\frac{1}{2}m\left(\left(\dot{x}_{D}-mb\sin\theta\dot{\theta}\right)^{2}+\left(\dot{y}_{D}+mb\cos\theta\dot{\theta}\right)^{2}\right)+\frac{1}{2}\left(\frac{1}{2}MR^{2}\right)\dot{\theta}^{2}+\frac{1}{2}\left(\frac{1}{2}mr^{2}\right)\dot{\varphi}^{2}$$

The longest one is the conservation of momentum... (with respect the origin of the coordinate systems)

$$\mathbf{L}=\left(x_{D}\dot{y}_{D}-y_{D}\dot{x}_{D}\right)M\hat{k}+\frac{1}{2}MR^{2}\dot{\theta}\hat{k}+\hat{k}\left(\frac{1}{2}mr^{2}\right)\dot{\varphi}^{2}+ m\left[\left(x_{D}\dot{y}_{D}+\dot{y}_{D}b\cos\theta+x_{D}b\cos\theta\dot{\theta}+b^{2}\cos^{2}\dot{\theta}\right)-\left(y_{D}\dot{x}_{D}-by_{D}\sin\theta\dot{\theta}+b\dot{x}_{D}\sin\theta-b^{2}\sin^{2}\theta\dot{\theta}\right)\right]\hat{k}$$

Could you tell me if these equations are correct? As a next step I am thinking of deriving each of the equations above and joining them in some way using the fact that $$\dot{E}=0$$,$$\dot{L}=0$$,$$\dot{p}=0$$. Is there an easier way?

4 years, 2 months ago

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