Check out the problem, which was to find the angle at which you can pull on the string from a spool without rolling the spool forwards or backwards.

If you pull at $\theta = 0$ (straight ahead), the spool will roll forward. If $\theta = \frac{\pi}{2}$, (straight up) your intuition should tell you the spool rolls backward. Somewhere in between, there’s a critical angle where the spool doesn’t roll at all.

Suppose we’re pulling at that critical angle, and the spool isn’t moving (so long as we pull lightly). There better not be any net torque on the spool.

You could go about mindlessly crunching away at this, which is what I originally did. But you can save yourself some effort by carefully choosing the axis about which you’ll measure the torque. There are two relevant forces on the spool – friction from the table and tension from the string. Choose the axis where the spool touches the table. That way, the torque from friction is automatically zero, and we just need to worry about torque from tension.

The torque from tension will vanish if the line along which the tension is applied intersects the axis. Geometrically, that looks like this:

To get the condition on $\theta$, slap a triangle on there whose sides you can figure out.

$\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) = \frac{r}{R}$

### 3 Responses to “Answer: Spool of String”

1. Nik Says:

Initially when I read the problem, I thought I would also consider the force of friction between the bit of string you’re pulling and the rest of the spool. But it would be negligible in comparison to the tension in the string, right?

2. meichenl Says:

We’re concerned with the motion of the spool, and so friction between the thread and spool is an internal force. There are many of those which we don’t consider. For example, the very top part of the spool is being pulled down by gravity, but the normal force from the table only acts on the bottom part of the spool. How then does the top part keep from falling down? The answer is that there are internal forces inside the spool itself. The spool is deformed somewhat from its ideal shape by these forces. In elastic theory, you would describe the deformation of the spool by a strain tensor, which would be related to a stress tensor by a certain differential equation. However, we’re treating the spool as a rigid body. That is, its bulk modulus and shear modulus are infinite. The speed of sound in the spool is infinite, and the coefficient of friction between the thread and the spool is infinite.

3. Nik Says:

I think you might have misunderstood me, I wasn’t talking about the internal forces. But anyhow, the idea was messed up. So never mind.