Physics Friday 100: Another Classic

Consider a spool of string with radius r, and ends capped by disks of radius R>r. The spool is placed on its side on a horizontal, so that it can roll on end rims. Suppose one pulls on an unwound end of the string, at some angle θ from the underside horizontal, as pictured below.

It’s clear from basic physical intuition that if θ=90°, so that one pulls straight up, the spool will tend to unwind, rolling to the left in our picture.

Similarly, if we pull horizontally, θ=0°, then the spool will roll to the right, and will wind up the string.

So, then, somewhere between these two angles should be a transition between these behaviors, a critical angle θc where the spool drags without rotating, neither winding nor unwinding. What is this angle?

There are four forces acting on the spool: the weight w of the spool, the normal force N of the surface on the spool, the force of friction f between the spool and surface, and the tension T of the string:
.
One could attempt to find equations relating these forces and solve for them, and thus the motion of the spool. However, there is a much simpler method. Let us consider the torque about the axis through the points of contact between the spool and the surface (point P in the figure above).
Both N and f act on the point P, so they produce no torque about this point. The weight w acts straight down, so the displacement vector and weight are antiparallel, and thus have a cross-product of zero, so the weight contributes no torque either.
Thus, the total torque about P is just that of the tension. We can see that a torque out of the plane of our drawing will mean rolling to the left, and thus unwinding, while a torque into the plane of our drawing will mean rolling to the right, and thus winding.

Therefore, our critical angle is that where the tension T exerts no torque. As T is non-zero, and the displacement vector r from P to the point on which T acts is always non-zero as well, the only way to have zero torque is if r and T are parallel. This means that T, and the unwound portion of the string, are along a portion of the line l tangent to the spool circle that passes through P.

Using trigonometry on the right triangle in the picture, we see that , and thus .
This problem serves as a classic example of how a little physical intuition and insight can greatly simplify a problem.

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