A ball is tied to a straw with a string. The ball is set to circle around the straw. As it rotates, the string winds up on the straw, and the string between the straw and the ball gets shorter and shorter. As this happens, the velocity of the ball increases.

Is velocity increasing only because r is decreasing in F=mv^2/r or is there another significant factor that is causing the velocity to increase?

(Also, can F be considered constant in this case?)


This question appears in Introduction to Classical Mechanics with Problems and Solutions by David Morin as problem 5.54. It is a problem that appears simple but is actually not so simple.

The first step is to realize that due to the non-negligible thickness of the pole, angular momentum is not conserved. However, there is another quantity conserved which is energy as the string does no work on the ball.

We next assume that each turn around the pole is very small. Each turn, three things will change. The height of the ball, the length of string remaining and the angle $\theta$ it makes with the pole. These three variables can be related using some geometry.

Applying conservation of energy, it can be shown that the following quantity is constant throughout the entire motion: $$v \sin \theta$$ where $v$ is the speed of the ball. Such a quantity is known as an adiabatic invariant.

$\theta$ will definitely increase as the length of the free string gets shorter and shorter. Since $0 \lt \theta \lt \pi/2$, $\sin\theta$ also increases and therefore (counterintuitively) $v$ actually decreases. You may be thinking about its angular velocity which does increase.


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