David SD's Comments

There's no contradiction with Galileo or Newton, and air resistance doesn't have anything to do with it.

Galileo says: in the absence of external forces, the center of mass of an object falls with an acceleration g, independent of the object's mass.

There are two problems here: 1) we should be looking at the acceleration of the center of mass of the board (which is halfway along, not at the end), and 2) there ARE external forces acting on the board (a slight upward force from the hinge).

Taking these facts into account is an elementary exercise in classical mechanics,* and the result is the following:

- the center of mass (middle) of the board moves with acceleration 3g/4. Note this is less than g because of the upward force from the hinge.
- the end of the board moves with twice the acceleration of the middle, 3g/2, which is greater than g, and therefore greater than the acceleration of the ball.

As an even more spectacular example, imagine that we attached a very light 50 foot stick to the board, with one end at the hinge. As the board falls, the other end of the stick would swing down at over 100 m/s. Is the end of the stick "falling" faster than the ball? No -- it's not free-falling. It's attached to a board. We need to take the whole setup into account before making random guesses about what will happen.

* Here's the calculation: the moment of inertia of the board is I=mL^2/3, where L is its length and m is its mass. The torque around the hinge is T=gmL cos(theta)/2, where g is the acceleration due to gravity, and theta is the angle between the board and the ground. For simplicity, Let's make the approximation that theta is small, so cos(theta) ~ 1. Then Newton's law says T=I d(theta)/dt, or d(theta)/dt ~ 3g/2L. The acceleration of the cup is then L d(theta)/dt = 3g/2.
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Reminds me of my favorite Carmina Burana interpretation: http://wookiessong.ytmnd.com/ The lyrics aren't as phonetically close to the originals as they are in the video above, but somehow they work... awesomely.
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  • Member Since 2012/08/07



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