Lab R3: The Plane Pendulum
Richard Martin
Department of Physics, Illinois State University
Abstract
We measured the period of a plane pendulum for various initial amplitudes in
order to check Galileo’s hypothesis that the period is independent of amplitude.
For angles less that about 37° we found the period to be constant at 0.90 ± 0.01
s, in agreement with the period prediction from the small angle approximation.
For larger initial angles the period increases as a function of initial amplitude.
1. Introduction
Galileo was the first to notice that the period of a pendulum is independent of its amplitude. It is
said that he did this by using his pulse to time the swinging of incense burners in church [1]. To
check Galileo’s result, we timed the period of a plane pendulum for 20 initial angles from a few
degrees up to about 90 degrees.
Figure 1. Diagram of the plane pendulum with bob, mass m, attached by massless
string, length L, making an angle θ with the vertical.
Figure 1 shows a schematic diagram of the pendulum, showing the pendulum angle θ, between
the pendulum string of length L and the vertical, and the pendulum bob of mass m. According to
the theory developed in class, the pendulum obeys Newton’s second law ma = F in the following
form:
d 2θ
mL 2 = −mg sin θ ,
dt
where g is the acceleration of gravity. Dividing by mL and putting both terms on the same side
of the equation, we get
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PHY 112 LAB
d 2θ g
+ sin θ = 0 .
dt 2 L
(1)
Equation (1) is called the differential equation of motion for the pendulum. If the angle θ
remains small, we can use the small angle approximation sin θ ≈ θ to rewrite (1) as
d 2θ g
+ θ ≈ 0.
dt 2 L
(2)
Equation (2) is the equation of motion for a simple harmonic oscillator [2], which has a solution
in terms of sines and cosines and a period
T = 2π
L
g
(3)
One of the goals of this experiment is to check Equation (3) and to determine the range of initial
angles for which it is valid. We will further plot the period as a function of initial angle to see
the relationship visually and to fit the part that deviates from the prediction of Equation (3).
2. Results and Discussion
Our apparatus was essentially that depicted in Figure 1. We used a ring stand with a horizontal
metal bar to support the pendulum. We had two strings attached to the pendulum at the same
point with their other ends attached to the horizontal bar at different points about 3 cm apart (the
two strings had the same length). This arrangement kept the bob swinging in a one plane. The
distance between the pivot and the pendulum bob was measured by a metal ruler to be L =
20.0 ± 0.3 cm, where the uncertainty was estimated by the three of us making independent
measurements and using half the spread as the uncertainty. The mass doesn’t enter into Equation
(3) but we measured it anyway, using a centigram balance, to be m = 402.2 ± 0.05 g, Thus, we
expect from Equation (3) that the period will be constant at
T = 2π
0.2 m
= 0.90 s
9.8 m/s 2
with an uncertainty of [3]
ΔT =
T ⎛ ΔL Δg ⎞
⎛ 0.003 0.005 ⎞
+
= 0.45 ⎜
+
⎟ = 0.007 s .
⎜
⎟
⎝ 0.2
2⎝ L
g ⎠
9.81 ⎠
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PHY 112 LAB
We measured the initial angles by measuring the perpendicular distance x from the vertical stand
to the bob and using trigonometry to get the angle:
⎛ x⎞
θ 0 = sin −1 ⎜ ⎟ .
⎝ L⎠
The period of the pendulum was determined by timing 10 swings and dividing by 10. We did 5
trials at each of 20 initial angles and used the standard deviation of the 5 measurements as the
uncertainty in the period. Table 1 shows one sample dataset, for initial angle x distance x = 0.01
m, or initial angle 2.9 degrees.
Trial
1
2
3
4
5
mean
SD
Period (s)
0.891
0.873
0.895
0.885
0.901
0.889
0.011
Table 1. Sample dataset used to determine the period of oscillation for initial angle 2.9 degrees.
Here 'mean' is the mean of the 5 measurements and 'SD' is the standard deviation, which is used as
an estimate of the uncertainty in the period.
We take the mean as the experimental value of the period, and the standard deviation as the
uncertainty, so this value is measured as T = 0.89 ± 0.01 s. We have truncated at the second
decimal place since the uncertainty is in that digit. Table 2 gives the periods measured in this
way for all the initial angles. You can see that, to within the uncertainty, the period agrees with
Equation (3) for initial angles smaller than 37 degrees. For angles larger than about 40 degrees,
Equation (3) does not yield the measured frequency within our estimated uncertainty. We have
no data points between 37 and 40 degrees so we cannot conclude anything in that region.
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PHY 112 LAB
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Initial angle
(degrees)
2.9
5.7
8.6
11.5
14.5
17.5
20.5
25.6
26.7
30.0
33.4
36.9
40.5
44.4
45.6
53.1
58.2
64.1
71.8
90.0
Period (s)
0.89 ± 0.01
0.90 ± 0.02
0.90 ± 0.01
0.89 ± 0.01
0.91 ± 0.02
0.91 ± 0.02
0.91 ± 0.02
0.91 ± 0.01
0.91 ± 0.02
0.9105 ± 0.01
0.92 ± 0.02
0.92 ± 0.02
0.925 ± 0.01
0.934 ± 0.01
0.94 ± 0.02
0.95 ± 0.02
0.96 ± 0.01
0.975 ± 0.01
0.995 ± 0.01
1.06 ± 0.01
Table 2. Data table showing the pendulum period for each of 20 initial angles.
The results for the period measurements are shown graphically in Figure 2. The dashed lines
define the uncertainty bounds on Equation 3, while the error bars show the uncertainty bounds
for each measured point. When these bounds overlap, we can say that Equation 3 correctly
describes the pendulum period. The graph shows visually what we noted from the data table,
that there is a clear division between the angles for which the small angle approximation holds
and the angles for which it fails. For initial angles greater than about 40 degrees there is a clear
increase in period as the amplitude increases.
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PHY 112 LAB
1.1
Period
Eq. 3 upper
Period (s)
1.05
Eq. 3 lower
1
0.95
0.9
0.85
0
20
40
60
80
Initial Angle (degrees)
Figure 2. Plot of the data from Table 2, showing the pendulum period as a function of initial angle.
3. Conclusions
In this lab we measured the period of a plane pendulum for 20 different initial angles. Since we
started the pendulum with no initial velocity, the initial angle is equal to the amplitude of motion.
The goal was to learn the behavior of the period and to check the approximate small angle
approximation theory. According to that theory, the period of a pendulum does not depend on
amplitude, being given by Equation 3. For the measured length of our pendulum, Equation 3
predicts a period of Tpredicted = 0.90 ± 0.007 s.
Our measured periods agree with this Tpredicted value only for amplitudes of about 37 degrees or
less. Initial angles above 40 degrees disagree with Equation 3 and do show a dependence on
amplitude. Figure 2 show this clearly as the data show the period increasing with initial angle
(amplitude) for angles greater that 40 degrees. Therefore, we can conclude that the small angle
approximation is not valid for amplitudes larger than about 40 degrees.
References
[1] R. F. MARTIN, PHY 380.03 lecture on the pendulum, 5 September 2010.
[2] B. K. CLARK, PHY 112 lecture on the simple harmonic oscillator, 3 March 2011.
[3] PHY 112 uncertainty analysis writeup: www.phy.ilstu.edu/~rfm/phy112lab/UncertaintyOutline.pdf
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