Sphere rolling inside cylinder - 3 dimensions

In summary, the problem involves a sphere rolling without slipping inside a hollow cylinder. The Lagrangian for this system includes motion in the z direction and can be expressed in terms of the angles of rotation for the sphere and the cylinder. The potential energy is given by U=-mg(R-r)cos\phi, while the kinetic energy includes both translational and rotational components.
  • #1
jimz
13
0

Homework Statement


A sphere of radius r and mass m rolls without slipping inside a hollow cylinder of radius R. z direction goes along axis of cylinder.
2irm5pl.jpg


Determine the Lagrangian with motion in the z direction included

Homework Equations


I let θ be the angle of the sphere rotation along the cylinder curve, φ be the angle from the cylinder center to the center of mass of the sphere, and ψ be the angle of rotation in z.

[tex](R-r)\theta=r\phi[/tex]
[tex]I=\frac{2}{5}mr^2[/tex]
[tex]z=r\psi[/tex]


The Attempt at a Solution


PE is easy.
[tex]U=-mg(R-r)cos\phi[/tex]

KE is harder... I think the translational KE is
[tex]\frac{1}{2}m[(R-r)^2\dot{\phi}^2+\dot{z}^2][/tex]

The rotational KE is troubling me... I want to say
[tex]\frac{1}{2}I(\dot{\theta}^2+\dot{\psi}^2)[/tex]
but I don't think that is right.

Any help would be great! Thanks.
 
Last edited:
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  • #2


Hello,

Your attempt at the solution looks good so far. However, for the rotational kinetic energy, you need to consider both the motion of the sphere along the cylinder (which is already accounted for in your translational KE) and the rotation of the sphere about its own axis. This can be expressed as:

\frac{1}{2}I(\dot{\theta}^2+\dot{\psi}^2)+\frac{1}{2}mr^2\dot{\phi}^2

The first term is the rotation of the sphere about its own axis, while the second term is the rotation of the sphere along the cylinder.

Hope this helps!
 
  • #3


Hello, as a scientist, I would like to provide some insight on your question. First, it is important to note that the Lagrangian is a function that describes the dynamics of a system in terms of its generalized coordinates and their derivatives. In this case, we have three generalized coordinates: θ, φ, and ψ. Therefore, the Lagrangian will be a function of these coordinates and their derivatives.

To determine the Lagrangian with motion in the z direction included, we need to consider the kinetic and potential energies of the system. Let's start with the potential energy. As you correctly stated, the potential energy can be written as U=-mg(R-r)cosφ.

Now, for the kinetic energy, we need to consider the translational and rotational motion of the sphere. The translational kinetic energy is given by T_trans = ½m[(R-r)^2φ̇² + ż²]. However, for the rotational kinetic energy, we need to consider the moment of inertia of the sphere about its center of mass. This can be written as I_cm = 2/5mr². But since we are dealing with a rolling motion, we need to consider the parallel axis theorem, which states that the moment of inertia about any axis parallel to the axis through the center of mass is equal to the moment of inertia about the center of mass plus the mass times the square of the distance between the two axes. In this case, the axis of rotation is passing through the center of mass of the sphere, so the moment of inertia can be written as I = I_cm + mr².

Therefore, the total rotational kinetic energy can be written as T_rot = ½(Iθ̇² + ψ̇²). Combining the translational and rotational kinetic energies, we get the total kinetic energy as T = ½m[(R-r)²φ̇² + ż²] + ½(Iθ̇² + ψ̇²).

Now, we can write the Lagrangian as L = T - U. Substituting the expressions for T and U, we get L = ½m[(R-r)²φ̇² + ż²] + ½(Iθ̇² + ψ̇²) + mg(R-r)cosφ.

I hope this helps you in determining the Lagrangian for this system.
 

1. How does the size of the sphere and cylinder affect the rolling motion?

The size of the sphere and cylinder can affect the rolling motion in several ways. A larger sphere will have more momentum and may roll faster, while a smaller sphere may roll slower. The diameter of the cylinder will also impact the rolling motion, as a larger cylinder may allow for more space for the sphere to move around inside, while a smaller cylinder may result in the sphere bouncing off the walls.

2. What is the relationship between the mass of the sphere and the speed of the rolling motion?

The mass of the sphere can impact the speed of the rolling motion. According to Newton's second law of motion, the acceleration of an object is directly proportional to its mass. Therefore, a heavier sphere will require more force to accelerate and will likely roll slower than a lighter sphere.

3. Can the direction of the rolling motion change inside the cylinder?

Yes, the direction of the rolling motion can change inside the cylinder. This can happen if the sphere collides with the walls of the cylinder at a certain angle, causing it to change its direction of motion. The shape and size of the sphere and cylinder, as well as the speed of the rolling motion, can also affect the direction of the motion.

4. How does friction play a role in the rolling motion of the sphere inside the cylinder?

Friction can have a significant impact on the rolling motion of the sphere inside the cylinder. Friction between the sphere and the walls of the cylinder can slow down the rolling motion, while friction between the sphere and the surface it is rolling on can also affect its speed. The amount of friction depends on the materials and surfaces involved, as well as the speed and direction of the rolling motion.

5. Can the rolling motion inside the cylinder be affected by external forces?

Yes, external forces can affect the rolling motion inside the cylinder. For example, if a force is applied to the cylinder, it can cause the sphere to roll faster or slower depending on the direction and magnitude of the force. Additionally, any external factors such as air resistance or vibrations can also impact the rolling motion of the sphere inside the cylinder.

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