B 01 CENCO Compound Pendulum Manual

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THE COMPOUND PENDULUM
OBJECT: To study the properties of a compound pendulum,
and to determine the acceleration due to gravity by the use
of such a pendulum.

and the application of Newton's second law of motion for
rotating bodies

METHOD: An experimental pendulum is suspended
successively about several axes at different points along its
length and the period about each axis is observed. A graph
is plotted of the period versus the distance of the axis of
suspension from one end of the pendulum. The nature of the
graph shows the physical properties of the compound
pendulum. From values of the period and the corresponding
length of the equivalent simple pendulum as determined
from the graph, the acceleration due to gravity is calculated.
From the mass of the pendulum and its radius of gyration as
obtained from the curve, the rotational inertia of the
pendulum is computed.

where
is the angular displacement,
the angular
acceleration, L the torque and I the rotational inertia of the
body.
Figure 1 represents a compound pendulum of mass m,
consisting of a rectangular bar AB to which a cylindrical

L

Ia

THEORY: A simple pendulum consists of a small body
called a "bob" (usually a sphere) attached to the end of a
string the length of which is great compared with the
dimensions of the bob and the mass of which is negligible in
comparison with that of the bob. Under these conditions the
mass of the bob may be regarded as concentrated at its
center of gravity, and the length l of the pendulum is the
distance of this point from the axis of suspension. When a
simple pendulum swings through a small arc, it executes
linear simple harmonic motion of period T, given by the
equation

T

2

l
g

(1)

where g is the acceleration due to gravity. This relation-ship
affords one of the simplest and most satisfactory methods of
determining g experimentally.
When the dimensions of the suspended body are not
negligible in comparison with the distance from the axis of
suspension to the center of gravity, the pendulum is called a
compound, or physical, pendulum. Any body mounted upon
a horizontal axis so as to vibrate under the force of gravity is
a compound pendulum. The motion of such a body is an
angular vibration about the axis of suspension. The
expression for the period of a compound pendulum may be
deduced from the general expression for the period of any
angular simple harmonic motion

T

2

a

Copyright, 2003, Sargent-Welch Scientific Company

(2)

mass M is attached. The pendulum is suspended on a
transverse axis through the point S. In the diagram, the
cylindrical mass M is represented as being exactly in the
middle of the bar, thereby making a symmetrical system in
which the center of gravity G is at the geometrical center.
Obviously, this particular condition is a very special case,
and has nothing whatever to do with the general treatment of
the problem.
In the equilibrium position, the center of gravity G is vertically
below the axis of suspension S. When the body is rotated
through an angle , the weight of the system mg, which
may be regarded as concentrated at the center of gravity,
exerts a restoring torque about S given by

L

mgh sin

(4)

No. 71990-154

where h is the distance from the axis of suspension to the
center of gravity. If a minus sign is used to indicate the tact
that the torque L is opposite in sign to the displacement ,
Eqs. (3) and (4) yield

Ia

mgh sin

suspension (as specified by h). Since the radius of gyration
of any given body is a constant, the period of any given
pendulum is a function of h only. Comparison of Eq. (1) and
Eq. (11) shows that the period of a compound pendulum
suspended on an axis at a distance h from its center of
gravity is equal to the period of a simple pendulum having a
length given by

(5)

l

When the angular displacement
is sufficiently small, sin
is approximately equal to
measured in radians. With this
restriction Eq. (5) may be written

mgh
I

a

(6)

k

(7)

where k is a constant. Equation (7) is the defining equation
of angular simple harmonic motion, i.e., motion in which the
angular acceleration is directly proportional to the angular
displacement and oppositely directed. Since the system
executes angular simple harmonic motion, substitution of the
expression for a from Eq. (6) in Eq. (2) yields the equation
for the period of a compound pendulum

2

T

I
mgh

h

h

ko 2
h

(12)

The simple pendulum whose period is the same as that of a
given compound pendulum is called the "equivalent simple
pendulum."
It is sometimes convenient to specify the location of the axis
of suspension S by its distance s from one end of the bar,
instead of by its distance h from the center of gravity G. If the
distances s1, s2 and D (Fig. 1) are measured from the end A,
the distance h1 must be considered negative, since h is
measured from G. Thus, if D is the fixed distance from A to
G, s1 = D + h1, s2 = D + h2 and, in general, s = D + h.
Substitution of this relationship in Eq. (11) yields

Since m, g, h and I are all numerically constant for any given
case, Eq. (6), may be written simply

a

h2

ko2

T

2

ko 2 s D
gs D

2

(13)

The relationships between T and s expressed by Eq. (l3) can
best be shown graphically. When T is plotted as a function of

(8)

where I is the rotational inertia of the pendulum about the
axis of suspension S. It is convenient to express I in terms of
I0, the rotational inertia of the body about an axis through its
center of gravity G. If the mass of the body is m,

Io

mko

2

(9)

where ko is the radius of gyration about an axis through G.
For any regular body, ko can be computed by means of the
appropriate formula (see any handbook of physics or
engineering); for an irregular body it must be determined
experimentally. The rotational inertia about any axis parallel
to the one through the center of gravity is given by

I

Io

mh

2

(10)

where h is the distance between the two axes. Substitution
of the relationships of Eqs. (9) and (10) in Eq. (8) yields

T

2

ko 2 h 2
gh

s, a pair of identical curves SPQ and S'P'Q' are obtained as
illustrated in Fig. 2. (The dotted portions represent
extrapolations over apart of the body where it is difficult to
obtain experimental data with this particular pendulum.)
Analysis of these curves reveals several interesting and
remarkable properties of the compound pendulum.
Beginning at the end A, as the axis is displaced from A
toward B the period diminishes, reaching a minimum value
at P, after which it increases as S approaches G. The two
curves are asymptotic to a perpendicular line through G,
indicating that the period becomes infinitely great for an axis
through the center of gravity. As the axis is displaced still

(11)

This equation expresses the period in terms of the geometry
of the body. It shows that the period is independent of the
mass, depending only upon the distribution of the mass (as
measured by ko) and upon the location of the axis of

2

Thus

farther from A (to the other side of G), the period again
diminishes to the same minimum value at a second point P',
after which it again increases.
A horizontal line SS', corresponding to a chosen value of T,
intersects the, graph in four points, indicating that there are
four positions of the axis, two on each side of the center of
gravity, for which the periods are the same. These positions
are symmetrically located with respect to G. There are,
therefore, two numerical values of h for which the period is
the same, represented by h1 and h2 (Figs. 1 and 2). Thus, for
any chosen axis of suspension S there is a conjugate point
O on the opposite side of the center of gravity such that the
periods about parallel axes through S and O are equal. The
point O is called the center of oscillation with respect to the
particular axis of suspension S. Consequently, if the center
of oscillation of any compound pendulum is located, the
pendulum may be inverted and supported at O without
altering its period. This so-called reversibility is one of the
unique properties of the compound pendulum and one that
has been made the basis of a very precise method of
measuring g (Kater's reversible pendulum).
It can be shown that the distance between S and O is equal
to l, the length of the equivalent simple pendulum. Equating
the expressions for the squares of the periods about S and
O, respectively, from Eq. (11)

T2

4
g

2

ko

2

h1

4
g

h1

from which

T

or

2

h1 h2

g

T

ko

2

h2
h2

ko

(20)

Substituting in Eq. (12) yields

lo

2ko

(21)

Thus, the shortest simple pendulum to which the compound
pendulum can be made equivalent has a length lo equal to
twice the radius of gyration of the body about a parallel axis
through the center of gravity. This is indicated in Fig. 2 by
the line PP'. Inspection of the figure further shows that, of the
two values of h for other than minimum period, one is less
than and one greater than ko. From the foregoing it is evident
that if two asymmetrical points S and O can be found such
that the periods of vibration are identical, the length of the
equivalent simple pendulum is the distance between the two
points, and the necessity for locating the center of gravity is
eliminated. Thus, by making use of the reversible property of
the compound pendulum, a simplicity is, achieved similar to
that of the simple pendulum, the experimental
determinations being reduced to one measurement of length
and one of period.
APPARATUS: The apparatus used in this experiment is very
simple, consisting merely of a rectangular steel bar
approximately 1 meter long carrying a heavy cylindrical
mass, and supported on a horizontal axis (Fig. 3). The bar

2

(14)

2

4

2

2

ho

h1

2

h2
g

(15)

(16)

Comparison of Eqs. (1) and (16) shows that the length l of
the equivalent simple pendulum is

l

h1

h2

(17)

Thus, the length of the equivalent simple pendulum is SO
(Figs. 1 and 2).
S' and O' are a second pair of conjugate points
symmetrically located with respect to S and O respectively,
i.e., having the same numerical values of h1 and h2.
Further consideration of Fig. 2 reveals the fact that the
period of vibration of a given body cannot be less than a
certain minimum value To, for which the four points of equal
period reduce to two, S and O' merging into P, and S' and O
merging into P' as h1 becomes numerically equal to h2. The
value of ho corresponding to minimum period can be
2
deduced by solving Eq. (14) for k0 , which yields

ko
and setting

ho

2

h1h2
h1

h2

Fig. 3. Experimental Pendulum

has a series of holes distributed along its length to provide
several axes of suspension. In use the pendulum is
supported successively at the various holes on a hardened
steel knife-edge in a wall bracket, and its period of vibration
determined with the aid of a stopwatch. A meter stick and a
platform balance with a set of weights are the only other
apparatus required.
PROCEDURE:
Experimental: Support the pendulum on the knife-edge at
the hole nearest to one end of the bar, making sure that it
swings freely in a vertical plane. With the aid of a stopwatch,

(18)
(19)

3

observe the time of 50 full vibrations and determine the
period. In making this determination, begin with the count of
"zero" as the pendulum passes through its central position,
count "one" as it makes its next transit through center going
in the same direction, etc. In a like manner determine the
period about an axis through each of the several holes.
Remove the pendulum from its support and with a meter
stick (preferably one equipped with caliper jaws) measure
the distances s1, s2, etc., of the various points of suspension
from one end of the bar. Record these lengths opposite the
corresponding periods.
Weigh the pendulum on the platform balance and record its
mass m.
Analysis of Data: 1. Plot the data in a graph similar to Fig.
2. Draw any horizontal line SS'. From the corresponding
period T as determined by the ordinate of this line, and the
length l of the corresponding equivalent simple pendulum as
given by the average of the values of SO and S'O', calculate
the acceleration g due to gravity, by means of Eq. (1) .
Compare with the accepted value and record the percentage
difference.
2. From the mass m of the pendulum and the radius of
gyration ko as determined from the graph, compute the
rotational inertia Io about the axis G by Eq. (9). Compute the
rotational inertia I about the axis S by Eq. (10).
QUESTIONS: 1. What is the minimum period with which this
pendulum can vibrate? What is the length of a simple
pendulum having the same period?
2. Describe how Fig. 2 would be altered if the cylindrical
mass M were near one end, say the end B.
3. With a given, axis of suspension, say S, discuss the
effect upon the period of (a) increasing the mass of the
cylindrical body; (b) moving it nearer to S.
4. How would the value of the minimum period To be
affected by moving the mass M in either direction from the
middle?
5. With the mass M near the end B and the pendulum
suspended about an axis S near A, how could the vibration
of the system about the axis S' be experimentally observed?
6. Does the center of oscillation of a solid body, such as a
rod or bar, lie within the body for any transverse axis of
suspension? Explain.
7. Locate the center of oscillation of a meter stick
suspended about a transverse axis at the 10cm mark. At
what other positions could the meter stick be suspended and
have the same period?
8. Prove that the period of a thin ring hanging on a peg is
the same as that of a simple pendulum whose length is to
the diameter of the ring.

4



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Title                           : Compound Pendulum Expt Manual CENCO
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