Ring, collar, and roller problem

[sdavidmiller]

Hey,
It's been about a year since I took mechanics, and I had agreed to help a friend out today. She gave me this problem, and I'm drawing a blank at how to approach it. No solution necessary, just an approach!

Homework Statement

A thin, uniform ring of mass m and radius R is attached by a frictionless pin to a collar at A and rests against a small roller at B. The ring lies in a vertical plane, and the collar can move freely on a horizontal rod and is acted upon by a horizontal force P. Express the angle THETA between the collar and the vertical axis at equilibrium

http://www.filedropper.com/second/173c0fdd1fddaa5be752d1f1324efcd1.jpg
EDIT: The image is at http://www.filedropper.com/wheelproblem It looks like it's having trouble showing up here, at least on my browser.

The Attempt at a Solution

Since this is a statics problem, I'm assuming all that is necessary is to set up equations so that the sum of all Forces, and the sum of all Torques, are each zero. I've immediately got a problem with the forces being zero, since the only horizontal force I can see is that of P.

With respect to Torque, I've tried the problem by putting the axis at C (which takes gravity out of the picture leaving me with only force P), B (which gives me a nice torque due to gravity of Rmgcos[Theta] but P is nullified, since it is parallel to it's position relative to B), and A (which, of course, takes P out of the equation).

Clearly, I'm missing something conceptually, because in my current way of thinking about the problem, there is no equilibrium short of at [Theta] = pi/2 -- when it can't rotate anymore!

 

[lippyka]

I'm guessing that I need to set up a free body diagram and find the static equilibrium equation. The sum of all forces in the x and y directions should be zero, and the sum of all torques should be zero. For the forces, P will be the only horizontal force, so we can set the sum of forces in the x direction equal to 0:P - mgsin[Theta] = 0For the torques, the following equation should work: Rmgcos[Theta] = 0We can then solve for Theta from the equations above. Theta = arcsin(P/mg)

 

[baba89]

Hi there,

It's great that you're trying to approach this problem and help out your friend. I would suggest breaking down the problem into smaller parts and considering the forces and torques acting on each individual component (ring, collar, and roller).

For the ring, you can consider the forces acting on it as the weight of the ring (mg) and the normal force from the roller (N). Since the ring is in equilibrium, the sum of these forces must equal zero.

For the collar, the only force acting on it is the horizontal force P. Again, since the collar is in equilibrium, the sum of forces must equal zero.

For the roller, you can consider the forces acting on it as the normal force from the ring (N) and the force from the collar (P). Again, since the roller is in equilibrium, the sum of these forces must equal zero.

Now, for the torques, you can consider the ring and collar as one system and the roller as another. For the ring and collar system, the torque due to gravity (mgRcos(theta)) must be balanced by the torque due to the horizontal force P (Pd) where d is the distance from the point of rotation to the point of application of the force. For the roller, the torque due to the normal force from the ring (NR) must be balanced by the torque due to the force from the collar (Pd).

By setting up equations for the sum of forces and torques for each component, you should be able to solve for the angle theta. I hope this helps and good luck with the problem!