Classical Mechanics Puzzler

Here's the problem demonstrated in class, in which a spool is resting on a rough horizontal surface. When the string (wrapped around the spool) is pulled directly upwards with tension T, the spool rotates ccw and accelerates to the left. It is straightforward to understand in terms of forces and torques why this happens. The string produces a torque that tends to rotate the spool ccw. The frictional force opposes this rotation with a force to the left. The string's torque wins over the friction's torque. You can see why this must be so from Newton's second law, F = ma. The tension, weight and normal force are vertical forces that sum to zero. The only  horizontal force is the frictional force which accelerates the spool to the left.

The second case is bit tricky. The string is pulled horizontally, and the spool translates to the right. Essentially, the torque created by the string tries to rotate the spool ccw, just as it did in the first case. And again, the frictional force tries to oppose this rotation with a force to the left. However, in this second case, the string's tension is to the right! Newton's second law yields

As it turns out, the string's force wins! It is not necessarily obvious why this is so. It would be good practice to solve this problem and show that this is indeed the case. But that's not the puzzle.

There is a unique angle at which these competing forces are at a standoff. At this angle, the spool will not rotate. If the tension T is high enough, the spool will slide. But this is simply because the maximum static friction force is overcome. (If the spool was part of a rack and pinion arrangement and couldn't slip, then there would be no motion regardless of the value of T.)

Describe in words, the special condition required for which the spool cannot rotate? The key to answering this puzzle lies in your point of view. Consider the torque created by every force in both cases from the point of contact between the spool and table. From this, what can you conclude about the direction of the string for the special case in which rotation cannot occur?

Be sure to enter your full name. Enter your email address ONLY if you want the number of points awarded e-mailed to you. Responses should be brief but complete. You must click the SUBMIT button to submit your entry. Entries are due by noon on Monday of next week.