Rolling without slipping and frictional force

[gcolombo]

I am posed the following problem:

A spool rests on a horizontal surface on which it rolls without slipping. The middle section of the spool has a radius r and is very light compared with the ends of the cylinder which have radius R and together have mass M. A string is wrapped around the middle section so you can pull horizontally (from the middle section's top side) with a force T.

2.1 Determine the total frictional force on the spool in terms of T, r and R.


2.2 What is the condition for the linear acceleration of the spool to exceed T/M?

2.3 Which direction does the friction force point when the acceleration is less than T/M? Which direction does the friction force point when the acceleration is greater than T/M?

My work so far is this:

For part 1, we solve a system of equations:
F_net = M*a = T - F_fr
with F_fr the force of friction,
Torque_net = I*alpha = T*r + F_fr*R
since the torques from the tension and friction operate in the same direction, and
a = alpha*R.

First we find I = (1/2)(M)(R^2 + r^2).
Then we substitute a/R = alpha.
To get F_fr in terms of T, R, and r, I substituted the moment of inertia and the expression for alpha into the net torque equation. Solving that for M*a, I then substituted into the net force equation and solved for F_fr, which, after some algebra, ends in the glorious result:

F_fr = T(R - r) / (R + r).

The second part of the question is quite intriguing.

Suppose that a > T/M. Then M*a > T, but M*a = T - F_fr. This implies that F_fr < 0, which has bizarre implications: the first is that r > R in the formula above, and the second (even more stunning) is that friction is actually pointing in the same direction as the relative motion!

Is this supposed to happen? If so, can someone point me to an explanation of how this defies a supposed tenet of friction (friction opposes relative motion)?
Did I make a mistake in setting up the system?

Thanks for any help!

 

[tiger_striped_cat]

Just something you might consider. friction opposes relative motion, but what about the case when you push a block (or calculator across the table) without it moving. Isn't it true that even friction acts when it's not moving at all? So there is a lot built into the phrase "relative motion". How about when you walk? Which way does friction point? Well really push your feet into the ground hard. You're feet push backwards (try it, can't you feel the force your of your feet pushing backwards before you break static friction. But if friction also pointed in that direction, then what would keep you from slipping? What is going on is that friction actually is pointing forward when you walk. Don't confuse your the motion of your body with what the relative motion of your feet WOULD experience if they were to slip. This is the key. In this example and the spool example static friction isn't broken. So I think you have your frictional force in the wrong direction. It certainly does point in the same direction that the spool rolls, but it also points opposite the motion the spool would experience if it were to slip.

Hope this help (and sorry for the long winded answer)

 

[wais]

Your work so far seems correct and your approach to solving the problem is also correct. It is not a mistake that you are getting a negative value for the friction force when the acceleration exceeds T/M. This is because in this scenario, the spool is experiencing a net force in the direction of its motion, which means it is accelerating and therefore, rolling faster. In this case, the friction force is acting to slow down the spool and prevent it from slipping, hence it is in the same direction as the relative motion. This is not a violation of the principle of friction opposing relative motion, as the friction force is still acting to prevent slipping. It is just that in this case, the direction of the relative motion and the direction of the friction force happen to be the same.

To answer the third part of the question, when the acceleration is less than T/M, the friction force will be in the opposite direction of the relative motion, as it is acting to increase the spool's acceleration and prevent it from slipping. When the acceleration is greater than T/M, the friction force will be in the same direction as the relative motion, as it is acting to slow down the spool and prevent it from slipping.

Overall, it is important to note that in this scenario, the friction force is always acting to prevent slipping and ensure that the spool rolls without slipping. Its direction may just happen to be the same or opposite to the direction of the relative motion, depending on the spool's acceleration.

I hope this helps clarify any confusion and good luck with the rest of the problem!