Statue of Galileo outside of the Uffizi, Florence
image courtesy of Open Parachute

 

Galileo Galilei (1564-1642) noticed that objects experiencing uniformly accelerated motion produced an interesting pattern relating the number of time intervals through which they moves and the distance which they ultimately travelled.

 

image courtesy of San Diego State University, Department of Natural Sciences

 

The spacing between locations shown in the diagram above could be charted as

 

time	position	change in position
0	0
1	1	1 = 1-0
2	4	3 = 4-1
3	9	5 = 9-4
4	16	7 = 16-9
5	25	9 = 25-16

 

Notice that the displacement between consecutive positions increased by odd integers. Also, notice that the total distance travelled, or net displacement since the objects were moving in a straight-line, from t = 0 was proportional to t².

 

Although his time was not our present day definition of a second (he used his pulse, a water clock, and a pendulum) he was the first to define the concept of uniformly accelerated motion as Dv/Dt equaling a constant value. He had to test his definition indirectly since he could not measure an object's instantaneous velocity. He conducted his trials using an inclined plane on which bells were spaced at increasing odd-integer intervals. The bells assisted with his timing by providing an auditory alert of when his "hard, smooth, and very round bronze ball" arrived at each position on the ramp. The incline's enormous length (12 cubits, or roughly 5.5 meters) allowed him enough time to take accurate measurements. In the picture below, you can see a replica of Galileo's ramp.

 

image courtesy of Arbor Scientific

 

We know from our discussions, that average velocity is defined as the quotient of net displacement over total time.

 

We also know from Galileo's definition that when an object experiences uniform acceleration,

 

 

and that its average velocity over an interval of time equals the average of its initial and final velocities during that time interval.

 

 

Using these facts, along with the requirement that the ball be released from rest, we have the equations,

 

 

Thus, Galileo devised a method of measuring accelerations. Although he never changed his ramp's elevation, we realize that different angles of elevation will result in different values for Dt and different values for a using the same displacement, or length of ramp. 

 

 

In our experiment, we will model Galileo's ramp and his determination of the behavior for naturally occurring accelerated motion by using an inclined plane, a motion detector, and the LabPro software. As we conduct our experiment, we will gradually increase the incline's angle of elevation recording the height between the top of the table and the BOTTOM of the board with each set of trials.

 

 

Notice from the diagram above, that the sine of the incline's angle of inclination equals height/length, or h/L. This relationship will be important when we construct our graph.

 

 

 

You will now use EXCEL to plot a scatter plot of average acceleration vs sin(q). Save your file with the format LastnameLastnameLastnameRamps.xls and email it to Mrs. Colwell.