1
1. measurement
MATHEMATICAL PENDULUM
1. Aim of the measurement
The aim of measurement is to develop empirically (i.e. based on measurements) a
formula describing the relationship to compute the period of the oscillation.
A pendulum is a weight suspended from a pivot so it can swing freely.
When a pendulum is displaced from its resting equilibrium position, it is subject to a
restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to
oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period.
The simple mathematical (or gravity) pendulum is an idealized mathematical model of
a pendulum. This is a weight (or bob) on the end of a massless cord suspended from a
pivot, without any kind of friction. When given an initial push, it will swing back and
forth at a constant amplitude. Real pendulums are subject to friction and air drag, so
the amplitude of their swings declines.
2. Theoretical background
The swinging motion of the pendulum is described by the Newtonian
equation of motion. The body is accelerated by the tangential component of the gravitational force GT=G·sin(), where G=m·g. The equation of motion is given by
ma
T
 mg sin(  )
(1)
Here m[kg] stands for the mass, g=9.81 kg m/s2 denotes the gravitational acceleration,
aT is the tangential acceleration, and  is the initial displacement.
Note that (1) teaches us that the mass of the bob does not influence the motion.
However, to verify this theoretical result, we will measure pendula of the same length but
different masses and of the same mass but different lengths.
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It is known that for small initial angles, the motion of the pendulum approximates the
simple harmonic motion. It is also known from the secondary school, that the harmonic
motion can be imagined as a projection of the steady circular motion, thus their kinematic properties (acceleration, velocity, etc.) agree (see figure).
According to the figure and (1), in the extreme position of the pendulum  ex  the acceleration and the amplitude of the approximate harmonic motion can be written as:
a  a cp   r  g sin 
2
ex
 g
and A  L sin 
ex
ex
 L
 r.
ex
(2)
In (2) it was assumed that the extreme position is small (<5o), hence the approximation
s in  e x   e x
holds. Since the period of the steady circular motion and the harmonic mo-
tion are equal, the final expression for the period (using (2)) is obtained as:
T 
2

 2
r
 2
a cp
L
ex
g
ex
 2
L
.
(3)
g
During the measurement, we will develop the above equation experimentally. Thus we
start off as if (3) would not be known and we seek the relationship between the period
and the length in the form of
T  AL
b
,
(4)
where A and b are unknown parameters (with dimensions). Our aim is thus to find out
the values and dimensions of these parameters – based on our measurements.
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3. The measurement
3.a Measuring the period of the motion of pendulums with same mass but different
lengths.
3.b Measuring the period of the motion of pendulums with same length but different masses.
The supervisor will show the pendula with different masses and lengths. The initial angle
should not be more than 5 degrees. The overall time of 20 swings is to be measured. (The
counting starts with zero.) The 10 measuring results will be put into the first four columns of the table below.
Measurement
Computations
#
L
m
n
n*T
T
logL
logT
P
[-]
[m]
[g]
[-]
[s]
[s]
[SI]
[SI]
[1]
1
2
…
20
The last four columns are needed for the computations described in the next section.
4. Post-processing of the measurements
4.a. Computing the actual values of A and b
Recall that our aim is to determine the parameters in (4). By taking the logarithm of (4),
we have
log T  log A  b  log L
which, if we introduce the new variables
x  log L
,
and
(5)
y  log T
, is the equation of a sim-
ple line in the form of
y  a  b x
where
a  log A
,
. Thus by plotting the measured data in the ( log
(6)
L
, log
T
) co-ordinate sys-
tem, the parameters A and b can be determined graphically as follows.
The data points – if the measurements were carefully performed – should form a straight
line that can be drawn with a ruler. (Approximately the same number of data point
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should be on both sides of the line.) Now pick two points on the two ends of the line
(x1,y1) and (x2,y2) and compute the slope:
y 2  y1
B  b 
x 2  x1
.
(7)
Finally, the intersection of the line and the y axis gives the other parameter:
A  10
a
.
(8)
4. b. Determining the dimension of the parameters
The dimension of parameter b is easy, because as this parameter is an exponent (Lb), it
has to be dimensionless.
Concerning parameter A, by analysing the formula
T  A 
L
we conclude that the dimension of A has to be such that the dimensions on the two
sides of the above equation become the same. Thus, we have
 A 
T0 


L 

s

.
(9)
m
After a little guessing, one might assume that A has to be inversely proportional to the
square root of the gravitational acceleration:
A  P /
(10)
g
Note that the parameter P in (10) is assumed to be dimensionless. Now the dimensions on
the two sides of eq. (4) agree and (4) turns into
T  P 
1
b
L ,
g
and the single dimensionless parameter P can be computed based on a single measurement
5. Preparation questions
1. Define the mathematical pendulum and give the formula for its period.
2. Make a sketch of a mathematical pendulum. Which parameters influence the period of the mathematical pendulum out of the followings: length, mass, gravitational acceleration, shape of the bob, thickness of the string.
3. Describe the aim and process of measurement. What quantities are to be measured?
4. In what form is the period sought? How does one determine its parameters?