Two Disk problem

Consider a system composed of two plane disks, which we will designate asDD and dd. Their radii are RR and rr respectively (R>r)(R>r). The disk dd is fixed over the disk DD and a distance bb from the center, as in the figure. The disk DD can spin and move freely in a friction-free platform, while the other disk dd is fixed on a point of disk DD, but can spin freely without friction. No external forces act on the system. The masses of the disks are MM and mm respectively.

Demonstrate that the angular velocity of both disks is a constant, in other words: d2θdt2=d2φdt2=0\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:

px=(M+m)x˙Dmbsinθ˙θ˙ p_{x}=(M+m){\dot{x}}_{D}\dot{-mb\sin\theta}\dot{\theta}

py=(M+m)y˙D+mbcosθ˙θ˙ p_{y} = (M+m){\dot{y}}_{D}\dot{+mb\cos\theta}\dot{\theta}

Then the equation for the conservation of energy:

E=12M(x˙D2+y˙D2)+12m((x˙Dmbsinθθ˙)2+(y˙D+mbcosθθ˙)2)+12(12MR2)θ˙2+12(12mr2)φ˙2 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)

L=(xDy˙DyDx˙D)Mk^+12MR2θ˙k^+k^(12mr2)φ˙2+m[(xDy˙D+y˙Dbcosθ+xDbcosθθ˙+b2cos2θ˙)(yDx˙DbyDsinθθ˙+bx˙Dsinθb2sin2θθ˙)]k^ \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 E˙=0\dot{E}=0,L˙=0\dot{L}=0,p˙=0\dot{p}=0. Is there an easier way?

Note by Salvador M
5 years, 9 months ago

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