Rotational Motion Conceptual Question

[ndomer]

Homework Statement

A spool of wire rests on a horizontal surface as in Figure P10.87. As the wire is pulled, the spool does not slip at the contact point P. On separate trials, each one of the forces F1, F2, F3, and F4 is applied to the spool. For each one of these forces, determine the direction the spool will roll. Note that the line of action of F2 passes through P.

http://www.webassign.net/pse/p11-68.gif

Homework Equations

Concepts Only

The Attempt at a Solution

I actually have the answers, I simply do not understand how to get to them.

F1, F3, and F4 all pull perpendicular to the axis of rotation in a counter-clockwise direction. My logic says that that would create a counter-clockwise rotation and friction would push the ball to the left. This is true for F3 and F4, but not for F1 (which the key says moves right).

I think I understand F2, but because I don't understand the others I want to make sure my reasoning is sound. I believe the perpendicular component of the force would act in the same way I described above rolling the ball left, but the force component going through the axis of the center of mass would roll the ball to the right. Because the force is at a 45 degree angle (because it starts at point P), the components are equal and the ball won't move.

 

[haruspex]

The answers drop out straight away if you understand instantaneous centres of rotation. As a wheel rolls, at any given instant the wheel is rotating about its point of contact with the ground. That, after all, is the only point of the disc that is not moving at that instant. So consider the moment of the force about P.

 

[ndomer]

The force acting that moves the wheel horizontally is friction between the ground and wheel, so the moment of force would be frictional force times the larger radius.

The applied forces move the wheel angularly. The direction of the angular motion determines which way the friction vector points.

I believe that is all correct, but am confused on determining which way the angular motion is going given the direction of the forces.

 

[haruspex]

The force acting that moves the wheel horizontally is friction between the ground and wheel

Why not the pull on the wire?

The applied forces move the wheel angularly. The direction of the angular motion determines which way the friction vector points.

That would work if the centre of the spool were on a fixed axis, but it isn't. Did you understand my explanation using rotation about the point of contact? Can you get the right answers if you stop looking for an intuitive solution and just go with the algebra?

 

[ndomer]

I was having trouble thinking it was a fixed axis, but using the rotation about the point of contact makes much more sense now.

Thank you very much haruspex.