PhysicsLAB Lab
Galileo Ramps

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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.
The spacing between locations shown in the diagram above could be charted as
change in position
1 = 1-0
3 = 4-1
5 = 9-4
7 = 16-9
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. 
Our experiment
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.
What is the total length of your board in centimeters? 

Starting with an initial incline elevated close to 10 cm, release your cart from rest a minimum of 40 cm in front of the motion detector, allow it to roll completely down the incline recording its motion on Lab Pro. Please CATCH the cart when it reaches the end of the incline, do NOT allow it to crash to the floor! As you watch the cart's position-time graph "grow" you will see the cart's accelerated motion (a parabola). As you watch its velocity-time graph "grow" you will see a linear relationship during the same time interval.
Highlight the appropriate "linear" section of the velocity graph (which corresponds with the parabolic section of the position graph) and fit a regression line. Record its slope in your data table. Note from the slope's units, that it represents the cart's acceleration (m/sec/sec). Gather information for a total of three trials at this angle.
Now elevate your board by at least 5 cm, recording your new height, and release the cart from rest three more times. Remembering with each trial, to record the slope of the regression line for the velocity-time graph displayed on LabPro. You will do this for a total of 7 different heights.
height trial 1 trial 2 trial 3 average slope sin q
elevation (cm) (m/sec2) (m/sec2) (m/sec2) (m/sec2) h/L
Data Analysis
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.
List the members in your group in alphabetical order, last name first.

What was the slope of the trend line of your group's EXCEL graph? 

What was the y-intercept of the trend line of your group's EXCEL graph? 

What was the correlation coefficient of the trend line of your group's EXCEL graph? 

Using the equation of your group's trend line, extrapolate the ball's average acceleration when q = 90º. 

Did your data for the cart's average acceleration "fit" better when the angles were smaller at the beginning, or when they were larger at the end? Explain your choice.

If we could assume that the cart only translated or "simply slid" down the incline, your extrapolated acceleration value should be close to that of the acceleration due to gravity, or 9.8 m/sec2. Calculate a percent error for your extrapolated value against this accepted value for the acceleration of a freely falling body. 

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