Scantling and Ball
acceleration due to gravity - SHM - Galileo's Inclined Plane
What 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.
Figure 1. Scantling track
Figure 2. Scantling track cross section
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 1/2
inch ball bearing. If performed quantitatively (this may be done in section instead) supply a meter
rule, stop watch and lycopodium powder.
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
and
but the is a constant velocity term and a changing velocity term. We can replace both velocities with an
average constant velocity using Mean Speed Rule
where v1 and v2 are the initial and final velocities. This leads to
Making quantitative measurements with the demo is probably more suited to section than lecture. Rating***
1 The lower track section was manufactured by CENCO with an equal length inclined portion. The longer 3m inclined track was made at Harvard.