Falling Faster than 'g'

What it shows:

Allow a board to rotate under the force of gravity and the free end will accelerate at a rate greater than g. Relation between angular acceleration and linear acceleration seems to give free-fall paradox.

How it works:

If a board, held in a vertical position with one end resting on the table, is allowed to topple over, every part of the board moves on a circular path. The center of percussion of the board is the point that has the acceleration of a free falling particle along the path that it follows (it's located 2/3 of the length of the board from the point of rotation); all points beyond the center of percussion descend with accelerations greater than they would have if they were particles moving freely on their respective paths. Consequently, the board reaches a certain position below which the vertical component of the acceleration of its end point exceeds the acceleration of gravity. For a uniform board, this is at such an angle that cos2θ is greater than 2/3, or θ is less than 35 degrees (see Comments below). The demonstration illustrates this fact and can be used to interpret the observed backward buckling of the top of a falling chimney, which is caused, in part, by inertial reactions.

The board is 1 meter long and rotates about a hinged end. A small plastic cup is attached 85 cm from the hinged end. A plastic golf tee is attached to the free end and holds a golf ball or large ball bearing when the board is propped up at a 35 deg angle by a vertical stick. When the vertical stick is suddenly yanked out, the board falls, and the ball falls into the plastic cup. The action is quick and startling.

uniform board

uniform board


The angular acceleration of the falling board is given by

angular acceleration

The acceleration of the free end of the board in the downward direction is


Making the substitution, this becomes


Finally, we have