Proof involving a cylinder rolling off of a larger cylinder

In summary, the conversation discusses the problem of a uniform right-cylinder balanced on top of a fixed cylinder with a slight disturbance causing the rolling cylinder to leave the fixed one at a specific angle. The conversation includes attempts at solving the problem and a hint provided to use the criterion of a block sliding down a frictionless sphere. Eventually, the conversation concludes with a solution using the forces of gravity and centripetal force.
  • #1
Salterium
3
0
1. The problem:
A Uniform right-cylinder of radius a is balanced on top of a perfectly rough (so that only pure rotation occurs) fixed cylinder of radius b (b>a), the axes of the two cylinders being parallel. If the balance is slightly disturbed, show that the rolling cylinder leaves the fixed one when the line which connects the centers of both cylinders makes an angle of Arccos(4/7) with the vertical.


Relevant equations
I=1/2 ma^2


The Attempt at a Solution



I really have no idea how to start. I understand that the cylinder will leave the circle when the curvature of the circle changes faster than the cylinder falls, so I used x^2 + y^2 = b^2 => y=(b^2-x^2)^(1/2) => y'=-x/(b^2-x^2)^(1/2). Then I set out to find dy/dt and dx/dt because I know that dy/dx= (dy/dt)/(dx/dt).

the force of gravity, mg is always downward, so all of the acceleration in the x direction comes from friction and the normal force, N. In the y direction, -ma= mg-N cos([tex]\theta[/tex]) ([tex]\theta[/tex] the angle with the vertical) and in the x-direction, m d2x/dt2 = N Sin([tex]\theta[/tex]).

This approach is not only wrong (notice the lack of consideration of a frictional force which is in both directions), but it is getting far too difficult in a hurry. I have absolutely no clue how else you could even begin to solve it. Mechanics is NOT my subject...
 
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  • #2
Did you happen to solve the "When does a block sliding down a frictionless sphere leave the sphere" problem in Physics 101 (or whatever they called it in your school)? The block leaves the sphere when the normal force between the block and sphere goes to zero. That occurs when the 'centrifugal force'

[tex] F = m \frac{v^2}{r} [/tex]

is equal and opposite to the component of the gravitational force normal to the surface of the sphere.

I think that the same criterion operates here and will allow you to solve the problem. The problem is phrased in such a way that you only need to consider a rolling cylinder and as a result you don't need to worry about the frictional force.
 
  • #3
Thank you so much for that hint, it was exactly what I needed to get on track. In order to stay on the circular path, some force has to pull the cylinder towards the center of the big cylinder with force mv^2/r, so the only force that can do that here is gravity (so friction isn't really involved). So I set the force of gravity in the radial direction equal to the centripetal force, used conservation of energy to get rid of v, and solved for theta. Perfect. Thanks again, this one was giving me trouble.
 
  • #4
You're welcome. I'm happy to have helped.
 

Related to Proof involving a cylinder rolling off of a larger cylinder

1. What is the physical phenomenon involved in a cylinder rolling off of a larger cylinder?

The physical phenomenon involved in a cylinder rolling off of a larger cylinder is a combination of rotational and translational motion. The larger cylinder provides a surface for the smaller cylinder to roll on, and due to the force of gravity and the shape of the objects, the smaller cylinder will roll down the larger cylinder.

2. How does the radius of the larger cylinder affect the speed of the smaller cylinder?

The radius of the larger cylinder does not affect the speed of the smaller cylinder. The speed of the smaller cylinder is determined by the force of gravity and the shape of the objects, not the size of the larger cylinder. However, a larger radius of the larger cylinder may provide a longer surface for the smaller cylinder to roll on, resulting in a longer distance traveled by the smaller cylinder.

3. What is the relationship between the mass of the smaller cylinder and its acceleration?

The mass of the smaller cylinder does not directly affect its acceleration. The acceleration of the smaller cylinder is determined by the force of gravity and the shape of the objects, not the mass. However, a larger mass may result in a greater force of gravity, resulting in a faster acceleration of the smaller cylinder.

4. How does the shape of the objects involved affect the motion of the smaller cylinder?

The shape of the objects involved, specifically the larger cylinder and the smaller cylinder, affects the motion of the smaller cylinder. The smaller cylinder must have a circular shape in order to roll on the larger cylinder, and the larger cylinder must have a curved surface for the smaller cylinder to roll on. If the shapes are not compatible, the smaller cylinder will not be able to roll off the larger cylinder.

5. What other factors may affect the motion of the smaller cylinder rolling off of a larger cylinder?

Other factors that may affect the motion of the smaller cylinder rolling off of a larger cylinder include the surface of the objects, any external forces acting on the objects, and any friction between the objects. These factors may alter the speed, distance, or direction of the smaller cylinder as it rolls off the larger cylinder.

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