Measuring time with music - Galileo's experiment

Micro controllers
Vector graphics

How fast is free fall? During the middle ages it was believed that objects fall with a constant speed, but no measurements were done to confirm that was actually the case. Suppose that you actually want to measure how fast a ball falls, but don't have access to modern equipment. A first problem is that falling objects moves too fast to accurately determine its position during the fall. This problem can be solved by letting a ball roll down an inclined plane, instead of falling freely. In this way only part of the gravitational force will affect the speed of the ball, and that will slow the process down. Now the ball will move slowly enough for it to be possible to measure position with reasonable precision, but in order to measure speed you also need to measure time. Measuring time is the same thing as dividing time into equal parts, but how do you do that with fractions of a second precision, without access to a watch? You could use music. A musician can keep the beat with very high precision, and if the beat in a piece of music is just a small fraction of a second off it is possible to hear that. It has been suggested that Galileo used music in order to measure time in the experiments where he discovered that objects in free fall move with constant acceleration, and not constant speed (Drake, S., The Role of Music in Galileo's Experiments. Scientific American, p. 98, June 1975.).
If you attach adjustable metal bands on the inclined plane it is possible hear when the ball passes each band. If you also play music at the same time as the ball rolling you can move the metal bands in such a way that the ball passes over a metal band at each beat in the music. By measuring the positions of the metal bands it is possible to how the distance the ball has moved depends on time.

Simulation of Galileo's experiment

An inclined plane is shown at the top of the simulation applet below. Attached to the plane are eight black bands. When you press the restart button the ball will start moving and music will start to play. Each time the ball passes over a black band you will hear a click. The black bands can be moved by dragging and dropping them at the desired position. If you have problems hitting the bands themselves use the tabs at the bottom of the bands instead. The bands can not be moved through each other. Start by moving the first band so that the click coincides with the first beat. Keep on pressing restart and moving the black bands until each click coincides with a beat. When you are done adjusting the positions look at the graphs at the middle and bottom of the applet. The graph in the middle (marked with v) shows the speed of the ball as a function of time, while the bottom graph (marked with x) shows the position of the ball as a function of time. The graphs do not have any particular scale, instead the maximum value will always end up at the top of the graph. If you have carefully placed each of the bands so that the clicks coincides with the beats you will find that the speed graphs has become a straight line. This means that the speed of the ball increases by the same amount between each band, which in turn is the same as saying that the ball moves with constant acceleration.