Faster than g—the answer

For Experiment 1 the final number of coins depends on two factors, the acceleration of the rod downwards and the friction between the coins and the rod.

When the rod is released it falls in accordance with the equation Torque = Moment of Inertia × Angular Acceleration (the equivalent in rotational terms to Newton's famous F = ma). The initial linear acceleration downwards of the rod can be calculated using

$$\begin{align*}{\text{vertical acceleration}} & = {\text{Angular Acceleration}}\; \nonumber\\ & \quad\times \;{\text{Distance to pivot }}\left( {{r}} \right)\end{align*}$$

The torque about the end of a uniform rod is mgL/2 (L/2 being distance to the centre of the rod). Since the moment of inertia is 1/3mL² we can obtain the angular acceleration of 1.5g/L.

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Thus, the linear acceleration downwards of each point along the rod is 1.5g/L multiplied by r the distance from the pivot. This gives a value of r = 2/3L for the distance from the pivot when a = g. Further along the rod the acceleration is greater than g...nearer the pivot less.

Thus, coins at >2/3L will definitely leave the road and fall vertically.

Coins at <2/3L will stay on the rod (depending on the value of the friction) and could be deposited in the pot. A high value of friction provides sufficient horizontal force to keep the coins on the rod, but for a lower value coins will likely slip off before the rod reaches the vertical. Different surfaces bring fewer or more coins into the box. I leave a full analysis of that to others.

Of course, the angular acceleration is not uniform, but caused by the component of the weight perpendicular to the rod. So, the analysis above is only true when the rod is being held horizontally. A cos term is needed for other angles, and integration to get the full picture.

The key of the second experiment is to gently move the coins so they are held vertically against the side of the glass (figures 2 and 3) using your thumb and finger. The coins can then be brought rapidly together (figure 4) above the glass. The coins do not have sufficient time to fall below the fingers if this is done rapidly. The task is made easier to perform if there is significant friction between the coins and fingers as the coins are less likely to slip down into the glass, giving the experimenter an extra fraction of a second the bring the fingers together

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