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] (

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?

Note by Salvador M
4 years, 11 months ago

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