Repeating Galileo's Experiment: Gravity and Acceleration

Objective

To research the experiments Galileo performed to calculate the acceleration due to gravity.

Difficulty

Procedure: Medium
Concept: Hard

Concept

Ancient mathematicians had to perform all of their experiments without the aid of high-tech equipment. This generally means that the experiments they performed were fairly simple in nature. Galileo's gravity experiments were no exception. By rolling different balls down a ramp from various heights, he was able to discover that the length the ball rolled was directly proportional to the square of the time taken.
Now, here's the trick - since Galileo didn't know exactly how much gravity affected objects, he wasn't just checking his results against some formula. He actually had to find a relationship between distance and time himself. This is the angle that we'll be taking during this experiment, because it's a lot more interesting than an experimental proof of a formula we already know.

Materials

• Grooved ramp (e.g. smooth cardboard)
• Ball
• Measuring stick/tape measure
• Stop watch or water clock

Procedure

1. Roll the ball down the ramp and measure the time it takes to roll from top to bottom.
2. Repeat a statistically significant number of times (>3) for different lengths (half way up the ramp, quarter, etc.) and different ramp angles. Make sure the ramp angle is shallow enough for measuring the time, but not too shallow such that friction dominates. Record the time for each trial.
3. In case a stop watch is not accessible, measure the time with water dripping into a cup, measure volume (or mass) of water, time will be proportional to water volume/mass. (Galileo actually measured time with heartbeat!)
4. Remark: this experiment (as well as Galileo's experiment), is known to ignore the friction and the fact that the ball is spinning and therefore losing some potential energy. If there is access to an air track (e.g. physics lab), friction could be significantly reduced.

 

 
         

Analysis

• Holding the distance constant, plot the distance (d) vs. time (t) and then the distance (d) vs. time squared (t²).
• Holding the ramp angle constant, again plot the distance (d) vs. time (t) and then the distance (d) vs. time squared (t²).
• Expected outcome: if we plot d vs. t, we find a parabola; d vs. t² would give a straight line. (Why?)
• Using the formula d = 1/2 at², calculate acceleration for each trial. Does this number seem high or low? Why is that, do you think?
• Using the formula g = a / sin(i), calculate gravity from experimental data. How much error do you have in your experiment? What are some possible sources for error?

Extensions

Does the mass of an object affect its acceleration due to gravity (i.e. do heavy objects accelerate faster than light ones)? What effect would the mass of an object have on this experiment, if any? How about size? Try a ball or two with a different radius to find out!

Source::  http://sciencefair.math.iit.edu/projects/galileo/