Mechanics problem

A horizontal circular disc of mass $M$ is free to rotate about a vertical axis through a point on its rim. If a dog(or pig if you wish) of mass $m$ walks once around the rim, by what angle does the rim turn through? Given that $\frac{M}{m}=\frac{8}{3}$ .

#Mechanics

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Comments

A uniform circular disk of mass $M$ and radius $R$ is confined to the $xy$ plane. It rotates with no frictional losses about the $z$ axis through a point on its edge. Gravity is into the page, and is thus irrelevant to this situation.

A bead of mass $m$ slides along a friction-less track on the circumference of the disk. The angle $\theta$ is the angle between the positive $x$ axis and a line from the origin to the center of the disk. The angle $\phi$ is the angle between the positive $x$ axis and a line from the the center of the disk to the bead.

Coordinates and velocities of the bead:

$x = R \, cos \theta + R \, cos \phi \\ y = R \, sin \theta + R \, sin \phi \\ \dot{x} = -R \, sin \theta \, \dot{\theta} - R \, sin \phi \, \dot{\phi} \\ \dot{y} = R \, cos \theta \, \dot{\theta} + R \, cos \phi \, \dot{\phi}$

Kinetic Energy of the system:

$E = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) + \frac{1}{2} \Big(\frac{1}{2} M R^2 + M R^2 \Big ) \dot{\theta}^2 \\ = \Big (\frac{1}{2} m R^2 + \frac{3}{4} M R^2 \Big) \dot{\theta}^2 + \frac{1}{2} m R^2 \dot{\phi}^2 + m R^2 \, sin\theta \, sin\phi \, \dot{\theta} \, \dot{\phi} + m R^2 \, cos\theta \, cos\phi \, \dot{\theta} \, \dot{\phi}$

Since all of the mass is at the same height with respect to gravity, the system Lagrangian is equal to the kinetic energy.

$L = E$

Equations of motion:

$\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{\theta}}} = \frac{\partial{L}}{\partial{\theta}} \\ \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{\phi}}} = \frac{\partial{L}}{\partial{\phi}}$

Evaluating the equations of motion yields the dynamics:

$\Big(m + \frac{3}{2} M \Big) \, \ddot{\theta} + m(sin\theta \, sin \phi + cos\theta \, cos\phi) \, \ddot{\phi} = m (cos\theta \, sin\phi - sin\theta \, cos\phi) \dot{\phi}^2 \\ m(sin\theta \, sin \phi + cos\theta \, cos\phi) \, \ddot{\theta} + m \ddot{\phi} = m (sin\theta \, cos\phi - cos\theta \, sin\phi) \dot{\theta}^2$

Initial conditions:

$\theta_0 = 0 \\ \dot{\theta}_0 = 0 \\ \phi_0 = 0 \\ \dot{\phi}_0 \neq 0$

We are interested in the angle $\theta$ which has elapsed by the time $\phi - \theta = 2 \pi$ , meaning that the bead has made a full revolution around the disk circumference. For $M = \frac{8}{3} m$ , the simulation is not very well behaved for some reason (it doesn't converge very well as the time step is varied). I will post a problem later asking for the results for the $M = 5 m$ case, since this case is very well behaved.

Steven Chase - 2 years, 3 months ago

Yeah u r right sir , thx a lot.