I am posed the following problem:(adsbygoogle = window.adsbygoogle || []).push({});

My work so far is this: 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?

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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Rolling without slipping and frictional force

**Physics Forums | Science Articles, Homework Help, Discussion**