- #1

erfz

## Homework Statement

I am referring to this thread and question: https://www.physicsforums.com/threads/rotation-of-a-spool-about-rough-ground.295666/

Here is the problem, restated:

A cylindrically symmetric spool of mass ##M## and radius ##R## sits at rest on a horizontal table with friction. With your hand on a light string wrapped around the axle of radius ##r##, you pull on the spool with a constant horizontal force of magnitude ##T## to the right. As a result, the spool rolls without slipping a distance ##L## along the table with no rolling friction.

Find the friction force ##f## acting on the spool.

## Homework Equations

##\tau_{net} = I\alpha##

##\tau = Frsin(\theta)##

##F_{net} = Ma_{cm}##

##\alpha = a_{cm} / R##, when not slipping

## The Attempt at a Solution

I am fine with the general method of this problem.

I would just like to make sure I understand what Doc Al means by "the rotational and translational accelerations must have the same sign" when applying the rotation without slipping condition.

This is what I make of his statement:

Consider a cylinder rolling without slipping on a table to the right. This also means, by intuition, that it must have an angular acceleration clockwise. So, if we take the convention that right is positive, then we should also take any clockwise torque contributions to be positive in ##\tau_{net} = I\alpha##.

We can take the opposite conventions if desired (left is positive, counterclockwise is positive).

Is my interpretation correct?

Also, this problem (at least in my notes) does not come with a diagram. Doesn't the derivation from the link assume the T is applied at the bottom of the axle, as opposed to the top? If it were applied to the top, I get that ##Ma_{cm} = T + f## and ##\tau_{net} = I\alpha = Tr - fR##. Applying the no slip condition and rearranging, the friction force is $$f = T\frac{-I + MrR}{I + MR^2}.$$

Is that correct? Doesn't this mean that if you apply ##T## to the top of the axle, ##MrR > I##? For the uniform cylinder stated, ##I = 1/2 ~MR^2## so ##MrR > 1/2 ~MR^2## and ##r > 1/2 ~R##. Is this true?

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