The effects of length, mass, start angle and gravitational constant on the period of a mathematical pendulum were investigated in this lab. It was determined that only two of these, length and gravitational constant, always had an effect on the period, with one, start angle, having an effect only at angles larger than 20°. The mass had no effect on the period. It was determined that the length was proportional to the period squared and the gravitational constant was proportional to the inverse of the period squared. By combing the two results an overall relationship between the two variables was found to be in the form:
T
2 
39 .
2
L g
[eqn. 1]
Pendulums are used throughout science, engineering, entertainment, business, and other industries for many purposes. Galileo did his own experiments on pendulums in order to determine what physical factors controlled them. Galileo was from a time when there were not a lot of hi-tech instruments to help with measurement. He had to rely upon simple experiments and his own questioning mind to understand the world around him.
A mathematical pendulum consists of a rigid (does not stretch) massless (mass is very small compared to overall mass) string with a bob mass on the end. At rest the pendulum hangs vertically. When it is pulled back to some initial start angle and let go, the pendulum oscillates back and forth about the vertical position with a regular period. This seemingly simple motion is comprised of many variables. There is the length of the string, the mass of the bob, the angle that it is drawn back to, and the height that it is hooked up at. The planet the pendulum is on may even affect its period of motion. This lab will help to answer the question: What does the period depend on ?
Because there is more than one variable to be tested, separate experiments were set up to hold some of the variables constant while one was changed to see its affect on the period. In each experiment, the period of the pendulum was our dependent variable, and the variables (length, start angle, bob mass, and local gravitational constant) are varied in four separate experiments. The first three variables (length, start angle, bob mass) are tested physically in the lab, while the fourth variable (gravitational constant) was tested using a simulation
(http://phet.colorado.edu/en/simulation/pendulum-lab).
Lab 10: The Mathematical Pendulum
Honors Physics 001
Maloney
Page 1 of 5

A pendulum was set up consisting of a light string hanging from a physics table stand as shown in Figure 1. Because the periods are comparable to human reaction time, ten periods were measured using a stop watch and then divided by ten to get the period for one back and forth oscillation. Four separate variables (mass, length, start angle and gravitational constant) were tested. When one variable was being modified all the others were held constant throughout the trials.
Length . The pendulum was adjusted to four different lengths and set in oscillation with a constant mass of 0.5 kg and from a set start angle of 10° for ten periods and repeated four times for each length.
Mass . The first variable that was tested was how the mass affected the period of the pendulum. Masses of varying size were attached to the pendulum and the
Figure 1: Pendulum Setup length was adjust slightly to make sure the distance from the pivot point to the center of mass of the bob was constant. The final constant length was 1.5 m. The pendulum was then pulled back to a set start angle of 10° and set in motion. Ten periods were measured and the process was repeated four times for each mass.
Start Angle . The pendulum was adjusted to eight different start angles and set in oscillation with a constant mass of 0.5 kg and a constant length of 1.5 meters. The time for ten periods was measured and the process was repeated four times for each angle.
Local Gravitational Constant . Because we cannot change the local gravitational constant, a simulation was used to test out the effect the gravitational constant has on the period. The simulation at the Phet website labeled
Pendulum Lab ( http://phet.colorado.edu/en/simulation/pendulum-lab ) was used. A pendulum of 1.5 m in length and 0.5 kg in mass was set in motion from a constant start angle of 10° on the Earth, Jupiter and the Moon in the simulation.
Graphing . All the data was taken and analyzed to determine what factors affected the period of the pendulum by graphing the data until no correlation was found or a linear correlation was found. The Period was plotted versus each variable directly. If a non linear relationship was found (curve) the period squared was plotted against the variable and so on.
Lab 10: The Mathematical Pendulum
Honors Physics 001
Maloney
Page 2 of 5

Period vs. Length
Table 1. Length and Period
L (m) T (sec) T 2 (sec 2 )
0.0
0.5
1.0
1.5
2.0
2.5
0.0
1.41

0.012
1.96

0.014
2.45

0.009
2.83

0.011
3.17

0.012
0.0
1.98
3.84
6.00
8.01
10.05
Period data was plotted versus length and analyzed with LoggerPro, the results of which can be seen in Figure 2A and 2B. Initially the data seemed to have a period is proportional to the square of length relationship so a second plot was done of the period squared vs. the length. This plot was found to be linear with a correlation of 0.9998 and a proportionality constant of 4.027. According to this data the period of a pendulum is proportional to the square of the length according to the equation
T
2 
4 .
027 L [eqn. 2]
Period vs. Mass
Table 2. Mass and Period
Period data was compared against mass data and showed almost no m (g) T (sec) correlation. The period changed by less than 1% throughout all the measurements and had a standard deviation of less than 1%. These
200 2.45

0.012
400 2.46

0.015 results show that the mass has no affect on the period of a mathematical pendulum. This makes sense, as the restoring force of a pendulum is the force of gravity on the bob, and yes that force will increase as the mass
600 2.45

0.018 increases, but the bob will have a proportional increase in inertia (mass) which in the end will cancel out the effect of the stronger gravitational
800 2.44

0.011 force in determining the motion of the pendulum. Because of all this, it was determined that the mass of the bob has no effect on the period of a mathematical pendulum.
Lab 10: The Mathematical Pendulum
Honors Physics 001
Maloney
Page 3 of 5

Period vs. Start Angle
Table 3. Start Angle and Period

(deg)
5
10
15
20
30
40
50
60
T (sec)
2.46

0.011
2.46

0.014
2.46

0.013
2.47

0.013
2.50

0.016
2.54

0.014
2.57

0.012
2.64

0.015
The experiment involving the effect of the start angle on the period yielded some strange results. Initially it looked like the start angle had no effect on the period of the pendulum. From 5° to 20° the value of the period remained almost constant and was well within the range of error. However after 20° the length of the period began to rise and continued to rise throughout the rest of the trials. A solution describing the pattern above 20° could not be found and all that can be said is that the period slowly increases. We can say that for small angles
(under 20°), the start angle has no effect on the period of the pendulum. One reason that may account for this may have something to do with the sine equaling the tangent of the start angle at small angles, simplifying the underlying physics of the pendulum at these small angles where the effect of the start angle cannot be easily detected. At larger angles the sine and tangent are not equal, complicating the physics of the pendulum in which the effect of the start angle must be taken into account. We will have to limit our results to say that the period is not affected by the start angle for angles smaller than 20°, and the exact effect is indeterminate for larger angles.
Period vs. Gravitational Constant
The investigation concerning the effect of the gravitational constant on the period of the pendulum was done using a simulation and thus there is no standard deviation to measure.
Gravity was simulated on the Earth (g = 9.81 m/sec 2 ), the Moon
(g = 1.62 m/sec 2 ), on Mars (g = 3.73 m/sec 2 ), on Neptune (g =
11.67 m/sec 2 ) and on Jupiter (g = 24.79 m/sec 2 ).
Table 4. Gravitational Constant and Period g (m/sec 2 ) T (sec) 1/T 2 (1/sec 2 )
1.62
3.73
9.81
11.67
24.79
6.04
3.98
2.46
2.25
1.54
0.027
0.063
0.165
0.198
0.422
Lab 10: The Mathematical Pendulum
Honors Physics 001
Maloney
Page 4 of 5

Discussion
Conclusion
Lab 10: The Mathematical Pendulum
Honors Physics 001
Maloney
Page 5 of 5