Scantling and Ball

Ball rolling down tilted trough in oscillatory fashion yields acceleration. Also known as Galileo's Inclined Plane).

scantling and ballWhat It Shows

An ingenious experiment to measure acceleration due to gravity first performed by Galileo.

How It Works

The measurement of the acceleration of a ball down an inclined plane requires two parameters, namely distance traveled and time taken. Distance is reasonably straight forward, and Galileo devised an ingenious way of accurately measuring the time taken based on his knowledge of the pendulum.

The scantling track consists of a gully with circular arc cross section (Figure 1). If the track is horizontal and the ball released from the lip of the gully, it will oscillate side to side with a fixed period, equivalent to the swinging of a pendulum bob. When the track is inclined and the ball again released from the lip, it will roll down the incline as it oscillates from side to side, tracing a sine wave path. Because the two motions are orthogonal they are independent, and the repeating sine wavelength maps out equal time intervals. The distance traveled in each time interval (say each wavelength) shows an odd integer increase in distance, indicative of uniform acceleration. Also, because a zero incline gives zero acceleration (constant velocity), and an increasing angle of inclination gives larger values of uniform acceleration, you can (and Galileo did) infer an acceleration for a 90° inclination, i.e. the acceleration due to gravity.

The track is in two pieces 1 (Figure 2), made of wood and painted matte black. The gully has width 10 cm and depth 1.5 cm. To trace the path of the ball the track can be given a fine dusting of lycopodium powder.

Setting It Up

Mount on a bench with lab clamps/bars to incline the upper track at about 20°. Use a ½ inch ball bearing. If performed quantitatively (this may be done in section instead), supply a meter rule, stop watch and lycopodium powder.


Figure 1. Scantling Track.

Figure 2. Scantling Track Cross-Section
scantling and ball scantling and ball


Comments

An historically important experiment, the inclined plane also allows an introduction to medieval mathematics. The derivation of acceleration can be obtained from the Mean Speed Rule, also known as the Merton College Rule because of its development, using "word algebra" (instead of equations) by Thomas Bradwardine at Merton College, Oxford, in the 14th Century. The two equations we need are:

scantling and ball and scantling and ball

but there is a constant velocity term and a changing velocity term. We can replace both velocities with an average constant velocity using Mean Speed Rule:

scantling and ball

where v1 and v2 are the initial and final velocities. This leads to

scantling and ball

Making quantitative measurements with the demo is probably more suited to section than lecture.

1 The lower track section was manufactured by CENCO with an equal length inclined portion. The longer 3-meter inclined track was made at Harvard.