Inclined Plane Lab
Background
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In the seventeenth century Galileo performed experiments which disproved the Aristotelian concept of the motion of falling bodies. Specifically, Galileo challenged the premise established by Aristotle that a heavy object falls faster and reaches the ground earlier than a light object. Legend says that Galileo disproved this concept by dropping two spheres of different weights simultaneously from the top of the leaning tower of Pisa and observed that they reached earth at the same time. While historians question the validity of this account, Galileo did perform a similar experiment for objects rolling down an incline plane and extended the results to freely falling bodies.
Galileo was interested in more than disproving one specific result of the Aristotelian worldview; he was seeking a completely new description of motion. He proposed a bold new hypothesis of his own; that the motion of a falling body is determined by constant acceleration, and that this motion is not affected at all by the weight or mass of the body. This experiment is designed to examine Galileo's hypothesis and to test its validity by experimenting in much the same way that he did. |
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In looking at the motion of a falling body, we can really measure only two things easily; the distance x traveled (which we can measure with a meter stick), and the time of fall t (which we can measure with a clock). Therefore, it is desirable to be able to look at acceleration in terms of time and distance. To do this step-by-step, we start with velocity. Velocity is defined as the change in displacement divided by the change in time, or
Equation 1: v = Dx/Dt = (x - x0)/(t - t0)
Where x = final position, x0 = initial position, t = final time, and t0 = initial time
In making a measurement, we can always arrange our meter stick so that the initial position is zero, and we can start our clock at the beginning of motion so that the initial time is also zero. Inserting these values into equation 1, we get the average velocity, vavg, of the motion as
Equation 2: vavg = (x - 0)/(t - 0) = x/t
Once we have defined velocity in terms of x and t, we can simply carry this one step further to define acceleration as the "change in velocity for a given change in time".
Equation 3: a = Dv/Dt or a = (v - v0)/(t - t0)
This is the quantity that Galileo claimed is constant in the motion of falling bodies. to make consequences of these as simple as possible, we arrange our experiment so that we start our clock at the beginning of motion, t0 = 0, and we start the object falling from rest, v0 = 0. In this case equation 3 becomes
Equation 4: a = v/t or v = at
Although final velocity cannot be measured directly, it can be calculated by using the equation for average velocity, vavg
Equation 5: vavg = (v + v0)/2
By substituting this equation into our previous equation for average velocity we get
Equation 6: vavg = x/t = (v + v0)/2
If v0 = 0, this yields
Equation 7: x/t = (v + 0)/2 or v = 2x/t
Finally, by combining equations 4 and 7 we find that
Equation 8: v = at = 2x/t ð 2x/t = at ð x = 1/2at2
Now, if acceleration a is a constant, our final result says: the displacement of a falling body starting from rest is directly proportional to the time interval squared and is independent of the mass of the object.
Thus, if Galileo is correct, a graph of x vs. t2 should give a straight line with a slope of 1/2 a.
Because of the experimental difficulty encountered in determining the time interval for a freely falling body, Galileo chose the inclined plane because the acceleration of gravity would be "diluted", thus allowing him to make a more accurate determination of time intervals using the water clocks available to him. To make things easier for ourselves, we will use modern electronic timers instead. Galileo used a polished inclined plane with smooth balls. He released the balls from different positions along the inclined plane and timed their travel over given distances. After measuring the time intervals for various distances, Galileo discovered that the distance travelled by the balls and the square of the time of travel were indeed related by a direct proportionality.
He repeated this experiment for different angles of inclination and found for each angle he obtained a new proportionality constant. Galileo observed that the proportionality constants increased as the angle of inclination increased. He asserted that, when the angle of inclination approached 90 degrees, the constant of proportionality would approach 1/2 the acceleration of a freely falling body as a consequence of equation 8.
The importance of Galileo's work with the inclined plane is found in its proof that all falling bodies, regardless of weight, fall with the same uniform acceleration. This proof aided him in refuting the Aristotelian physics of motion and represented a triumph of the scientific method that Galileo played a great part in advancing.
Procedure
1. Practice starting the marbles and clock simultaneously a few times until you can time the marble's trip down the incline with some confidence. Start the marble rolling by setting it at some specified distance and holding it there with a pencil or ruler in front . You can start it by quickly removing the pencil. This will prevent you from accidentally giving the marble an extra push when you start it. Note that there will be a stopping block at the bottom of the inclined plane.
2. Place the marble on the inclined track at the distance specified on your data sheet.
3. When the marble is released, start your timer. When the marble strikes the stopping block, record the elapsed time in units of seconds.
4. Repeat your measurements for this distance two more times, recording the elapsed time in your data table.
5. Repeat the experiment for other distances in your data table.
6. Repeat the entire experiment for the lighter mass.
7. Plot graphs of distance versus average time for each marble.
8. Make graphs of distance versus average time squared for each marble.
Part a: Heavy Marble
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Distance (cm) |
Elapsed Time Trial 1 |
Elapsed Time Trial 2 |
Elapsed Time Trial 3 |
Average Time |
Average Time Squared |
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0 |
0 |
0 |
0 |
0 |
0 |
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20 |
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40 |
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60 |
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80 |
Part b: Light Marble
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Distance (cm) |
Elapsed Time Trial 1 |
Elapsed Time Trial 2 |
Elapsed Time Trial 3 |
Average Time |
Average Time Squared |
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0 |
0 |
0 |
0 |
0 |
0 |
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20 |
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40 |
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60 |
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80 |