Measurement of the Acceleration Due to
Gravity
Phys 303 Lab
Experiment 0
Justin M. Sanders
January 12, 2004
Abstract
Near the surface of the earth, all objects freely fall downward
with the same acceleration g called “the acceleration due to gravity”.
The acceleration due to gravity has been obtained from the period of
pendulums of various lengths, and it was found that in Mobile, AL
g = 9.7 ± 0.2 m/s2 .
1
1
Introduction & Theory
Near the surface of a planet, all objects experience the same acceleration due
to the force of gravity. In Mobile, Alabama, this acceleration is 9.793394 m/s.1
This constant acceleration due to gravity appears in the relationship of the
period of a simple pendulum to its length. Therefore, by measuring the period of a simple pendulum as a function of its length, the acceleration due
to gravity may be determined.
A simple pendulum consists of a point mass m suspended from a pivot
by a massless, unstretchable string of length l as shown in Figure 1. If, at
Figure 1: Schematic diagram of the simple pendulum.
some point in the pendulum’s swing, the string makes an angle θ with the
vertical, the torque on the pendulum is given by
τ = −mgl sin θ.
(1)
Using Newton’s Second Law for rotation, and taking the moment of inertia
of the simple pendulum to be I = ml2 , we obtain
d2 θ
= −mgl sin θ.
(2)
dt2
Rearranging and making the approximation sin θ ≈ θ for small angles results
in the differential equation for a simple harmonic oscillator
ml2
d2 θ g
+ θ=0
dt2
l
which has an angular frequency ω =
simple pendulum (for small angles) is
(3)
q
g/l. Therefore, the period T of a
s
T = 2π
l
.
g
(4)
If the period and length of a simple pendulum are known, Eq. 4 can be solved
for the acceleration due to gravity g:
g=
4π 2 l
.
T2
1
(5)
Physics Experiments for PH201 and PH202, (USA Dept. of Physics, Mobile, AL,
1998), p. 41
2
This will be referred to as Method A.
Alternatively, if the period T is plotted versus the square root of the
2π
. The slope can then be
length, a straight line should result with slope √
g
solved for g. This will be referred to as Method B. Both of these techniques
will be used in this work.
2
Procedure
A small metal annulus was suspended by a thin thread from a support which
allowed the length of the thread to be varied. The resulting pendulum was
drawn aside by an angle of approximately 10◦ (judged by comparison with
a paper template attached to the support) and released. Simultaneous with
the pendulum’s release, a stopwatch was started. The stopwatch (least count
0.01 s) was stopped when the pendulum had completed ten oscillations. The
length of the pendulum was measured using a standard meter stick (least
count 1 mm) between the point of support and the center of the metal annulus. The period of the the pendulum was taken to be the one-tenth of the
time required for ten oscillations.
For one pendulum length, the measurements were repeated ten times
in order to obtain and estimate of the variation of the period times. The
resulting standard deviation was used for the uncertainty in the period.
3
Data
For all length measurements, the uncertainty is taken to be 3 mm, since there
was some variation in determining the position of the point of support and
the center of the annulus.
For the 0.090 m length, the average period was 0.591 s and the standard
deviation was 0.005 s. This standard deviation was used as the uncertainty
in the period for all the lengths.
3
Length (m) Period (s) Calculated g (m/s2 )
0.387
1.231
10.082 ± 0.1
0.353
1.194
9.775 ± 0.1
0.268
1.031
9.954 ± 0.1
0.201
0.888
10.063 ± 0.1
0.174
0.818
10.266 ± 0.1
0.130
0.713
10.095 ± 0.1
0.090
0.587
10.312 ± 0.1
0.090
0.591
0.090
0.594
0.090
0.591
0.090
0.588
0.090
0.600
0.090
0.597
0.090
0.591
0.090
0.584
0.090
0.590
Table 1: Pendulum Lengths and Periods
4
4.1
Analysis and Results
Method A: Sample Calculation
Using Eq. 5:
(4π 2 )(0.387 m)
= 10.1 m/s2
(1.231 s)2
and propagating the errors
(6)
g=
δg = g
v
!
u
u δl 2
t
l
δT
+ 2
T
!2
(7)
and evaluating
s
δg = (10.1 m/s2 )
0.003 m
0.387 m
2
+ 2
0.005 s
1.231 s
2
= 0.1 m/s2
(8)
The results of these calculations are given in Table 1, and the average of the
values of g given in the table is 10.1 ± 0.1 m/s2 .
4
Figure 2: Plot of period versus square root of length for a simple pendulum.
The solid points are the present measurements, the dashed line is a leastsquares fit using Eq. 9
4.2
Method B
√
The data in Table 1 used to produce a plot of T is plotted versus l. The plot
is shown in Figure 2. A least-squares fit to the data was performed using the
fitting routine in the program GNUPLOT (ver. 3.71) and the model chosen
was a two-parameter straight line:
√
T =a l+b
(9)
The fit yielded the following values for the parameters: a = 2.0204±0.02486 s/m1/2
and b = −0.0171951 ± 0.0119 s. The slope a can be solved for g:
g=
4π 2
4π 2
=
= 9.6773 m/s2
1/2 2
a2
(2.0204 s/m )
(10)
and the uncertainty is
δg = g(2
0.02486
δa
) = (9.6773 m/s2 )(2
) = 0.238 m/s2
a
2.0204
(11)
Thus the result of Method B is that g = 9.7 ± 0.2 m/s2 .
5
Conclusions
In this work, the dependence of the period of the simple pendulum on its
length was exploited to obtain the acceleration due to gravity. The period of
the pendulum was determined by timing its swing for various lengths, and
two methods were used to analyze the data. In Method A, Eq. 5 was used for
each length to obtain a value for g, then all the g’s were averaged together
resulting in gA = 10.1 ± 0.1 m/s2 . Since
|g − gA |
|9.793394 m/s2 − 10.1 m/s2 |
=
= 3.1,
δgA
0.1 m/s2
5
(12)
the analysis using Method A disagrees with the “true” value of g. However,
by performing a linear fit in Method B, the deviation from the “true” value
was smaller:
|9.793394 m/s2 − 9.7 m/s2 |
|g − gB |
=
= 0.5,
δgA
0.2 m/s2
(13)
The primary difference between these two methods is that the parameter
b in Method B can account for systematic errors in the period. It has been
determined that time intervals measured by stopwatch from visual clues tend
to be biased toward short times.2 The effect of a bias toward short periods
in Eq. 5 would be to shift the calculated value of gA upward, and Method A
did indeed yield a high value of g. From Eq. 4, it can be seen that for the
0.387 m pendulum, the period should be
s
T = 2π
0.387 m
= 1.249 s
9.793394 m/s2
(14)
The time for ten periods would be 12.49 s, which means that the measured
interval of 12.31 s was 0.18 s short. This is comparable to the 0.14 s reaction
time found previously. The value of parameter b implies that the ten-period
measurements are consistently short by 0.17 s, and again this is consistent
with the reaction-time findings.
The results of these measurements show that the fitting method for obtaining g is superior to more straight-forward method (A), since it reduces
the effect of one type of systematic error.
2
J.M. Sanders, Measurement of Human Reaction Times, unpublished.
6