The Simple Pendulum: Finding Relationships
Purpose: The purpose of this experiment was to find out if length, mass, and angle of
swing (also called amplitude) affect the period of a simple pendulum.
Apparatus and Procedure: There were actually three experiments conducted.
1. In the first experiment the length (l) of the pendulum was the independent variable
and the period (T) was the dependent variable. The mass (m) was held constant at
66 g and the angle (A) of swing was held constant at 10° (see diagram of the
apparatus below).
Length of
pendulum
string
angle of
pendulum
Pendulum
bob
2. In the second experiment the mass (m) of the pendulum was the independent variable
and the period (T) was the dependent variable. The length (l) was held constant at
100 cm and the angle (A) of swing was held constant at 10°.
3. In the third experiment the angle (A) of the pendulum was the independent variable
and the period (T) was the dependent variable. The length (l) was held constant at
90 cm and the mass (m) was held constant at 66 g.
In each experiment one experimenter was responsible for releasing the pendulum
bob and counting 10 complete swings (i.e. back and forth 10 times). The time was
divided by 10 to determine the time for one swing (the period). Each time a variable
was varied, three trials were executed and the average period calculated. One
experimenter acted as timer using a stopwatch and signaled the beginning of the
timing. Another experimenter made sure the pendulum bob was released at the
appropriate angle as measured by a protractor. A meter stick was used to vary the
length of the pendulum by adjusting a pendulum clamp and raising and lowering a
string tied to the pendulum bob. (See diagram above.)
Raw Data:
Mass = 66 g
Angle = 10°
length Trial 1 Trial 2 Trial 3 Avg. Period
(cm) (s) (s) (s) (s)
25 .99 1.00 .98 0.99
40 1.24 1.26 1.25 1.25
55 1.48 1.49 1.50 1.49
70 1.69 1.67 1.68 1.68
85 1.83 1.83 1.83 1.83
100 2.01 2.03 2.02 2.02
Length = 100 cm
Angle = 10°
mass Trial 1 Trial 2 Trial 3 Period
(g) (s) (2) (s) (s)
5.5 1.98 2.00 1.99 1.99
23 2.01 1.99 2.00 2.00
66 1.98 1.98 1.98 1.98
130 2.01 1.99 2.00 2.00
* only 4 different masses were available; 5 different masses would have been preferable
Length = 90 cm
Mass = 66 g
Amplitude Trial 1 Trial 2 Trial 3 Period
(°) (s) (s) (s) (s)
5 1.88 1.90 1.89 1.89
10 1.88 1.89 1.89 1.89
15 1.92 1.90 1.91 1.91
20 1.93 1.89 1.91 1.91
45 1.91 1.92 1.92 1.92
Evaluation of Data:
This graph needed to be linearized.
It appears to be of the general form
y = k xn where “n” is less than
zero.
This graph did not need to be
linearized. It appears to be of the
general form y = k xn where “n” is
zero.
This graph did not need to be
linearized. It appears to be of the
general form y = k xn where “n” is
zero.
In order to linearize the period vs. length data, the length data had to be raised to the 0.5
power. As shown in the graph above, the graph became linear. From the regression line
and statistics calculated by the Graphical Analysis software an algebraic relationship
could be generated:
T = (.204 s/cm.5) l.5 - .0302 s
By using the 5% rule, the y intercept can be ignored since it is only 1.5% of the largest
period value. Therefore, the equation reduces to:
T = (.204 s/cm.5) l.5
The period of a simple pendulum is directly proportional to the square root of the length
of the pendulum. The slope has the units of s/cm.5 and its significance will be discussed
in the “Conclusion” section.
Conclusion:
Several new terms were introduced through this experiment. The period of a
simple pendulum is the amount of time required for the pendulum bob to swing back and
forth one time. This term can actually be used to describe any event that repeats in a
regular way such as the period of the second hand of a clock being 60 s and likewise the
period of revolution of the Earth around the sun is 365 days.
Another important term introduced through this experiment was linearization.
Linearization is the process by which the exact algebraic relationship between two
variables is determined by performing calculations on one variable until when graphed, a
linear relationship appears. This was evidenced in the case in this experiment when
initially period vs. length yielded a curved graph (not linear) but period vs. length to the
0.5 power yielded a straight graph.
Much was learned about the behavior of a simple pendulum. Length appears to
affect the period of a simple pendulum but mass and angle of release do not. In the post-
lab discussion, it was revealed that the pendulum’s behavior is different for small angles
in comparison to large angles. No precise relationship was discussed. Perhaps this would
be an appropriate topic for additional exploration.
A relationship between period and length was determined:
T = (.204 s/cm.5) l.5
In the post-lab discussion, it was revealed that the slope does indeed have significance. It
can be re-written:
T = 2 (l/g.5)
Where “g” is the acceleration of gravity for Earth = 980 cm/s/s. The equation can be
rearranged:
T = (2 /g.5) l.5
When evaluated, the term in parenthesis is found to equal 0.2. Therefore, the slope that
was found must equal the term in parenthesis, 2 /g.5. Since the results obtained in this
experiment are in agreement with a well-established relationship, they can be considered
to be good results.