Lagrangian mechanics: sphere inside a cylinder

[skeer]

The problem goes by this:

A sphere of radius $\rho$ is constrained to roll without slipping on the lower half of the
inner surface of a hollow cylinder of inside radius R. Determine the Lagrangian
function, the equation of constraint, and Lagrange's equations of motion. Find the
frequency of small oscillations.

What I am thinking:

Problem: I do not know how to relate the distance traveled by the center of mass of the sphere inside the cylinder.
The sphere has a constraint related to its center of mass and its diameter. If it was in a plane it would be that the distance traveled its equal to the radius times an angle: $x = \rho\times\theta$. However, the sphere is inside the cylinder...

The sphere is rolling, thus it has kinetic energy. My question is: should I used the time derivative of the angle that I used to relate the motion of the sphere ?

Any help would be appreciated. Thank you!

 

[Matterwave]

Homework problems belong in the homework section. :)

So your sphere is constrained to move along the cylinder's surface, that's 1 constraint on a 3-D problem, which means there's two ways you can go about solving the problem. You can pick 3 generalized coordinates and use Lagrange multipliers to solve the problem, or you might be able to simply do this problem with 2 generalized coordinates. The trick in a lot of these Lagrangian problem is to find a good set of generalized coordinates that make the problem simpler. What do you think are good candidates for the coordinates?

 

[skeer]

I am sorry for posting here. I will be more careful next time.

What do you think are good candidates for the coordinates?

The sphere can only move up and down the cylinder. I could describe this motion with cylindrical coordinates where my $r$ and $z$ are fixed leaving only $\theta$ to vary
The second type of motion is rotational. For this I would use polar coordinates. Here, my radius is fixed too, which leave another angle $\phi$ to vary.
Is this a good way of thinking about the general coordinates?

 

[Matterwave]

I am sorry for posting here. I will be more careful next time.The sphere can only move up and down the cylinder. I could describe this motion with cylindrical coordinates where my $r$ and $z$ are fixed leaving only $\theta$ to vary
The second type of motion is rotational. For this I would use polar coordinates. Here, my radius is fixed too, which leave another angle $\phi$ to vary.
Is this a good way of thinking about the general coordinates?

If you fix r and z, then there is just $\theta$ as measured by the cylinder. So you are considering the sphere only rolling up and down, and it can't roll forward or backwards?

What angle is $\phi$? How are you measuring this angle? Remember that the sphere is confined to roll without slipping. This is an important constraint.

 

[skeer]

If you fix r and z, then there is just $\theta$ as measured by the cylinder. So you are considering the sphere only rolling up and down, and it can't roll forward or backwards?

The problem said it would move on the lower half of the cylinder. I was thinking that it might only be able to move as it was in a big U. Now, would it matter if it can be moved forward and backward if we are to only look at their potential and kinetic energy? My thinking is it does not, but I am not sure.

What angle is $\phi$? How are you measuring this angle? Remember that the sphere is confined to roll without slipping. This is an important constraint.

Since the sphere is rolling without slipping, I was thinking that I could measure an angle $\phi$ from the cylinder to relate its translational motion to the rotational motion. Now, this angle might be also $\theta$ as measured by the cylinder.

Normally, if I roll a sphere without slipping, I can relate its translational motion to its rolling by $x= \phi \rho$ where $x$ is the horizontal distance, $\phi$ the angle that the sphere has turn, and $\rho$ its radius. In the case of the problem maybe the relation should look like $\theta R = \phi \rho$; thus relating the rotational motion to the translational motion along the arc. And maybe this is the equation of constrain that the problem ask for.

Using this, my equations would be: $$T = (1/2) m (R\dot{\theta} / \dot{\phi})^2 + (1/2) I (\dot{\phi})^2 $$ $$U = (R -(R-\rho)cos(\theta))mg$$

 

[Matterwave]

If you do not allow motion along the z-axis, then the problem should be entirely determined by one variable since you've reduced a 3-D problem into a 1-D problem with 2 constraints (no motion along z, and rolling without slipping). You should not have a need for both $\theta$ and $\phi$ in doing your problem. Your constraint looks fine to me, you can use it to get rid of one of the variables though.

Also, check your $T$ since the first term doesn't appear to have the right units.

 

[skeer]

I double checked my $T$, my constraint, and my potential. I correct them taking into account that I would measure the motion of the center of mass. I think that with these corrections and your help, I was able to figure out the problem.

Thank you!

P.S. My potential and kinetic energies were the following:

$T = (1/2) I \dot \phi ^2 + (1/2) (\dot x ^2 + \dot y^2)m$ where $x= (R-\rho)sin(\theta)$ and $y = R - (R-\rho)cos(\theta)$

$U= [R-(R-\rho)cos(\theta)]mg$

The constraint equation to eliminate $\phi$ was $(R-\rho)\theta=\rho\phi$