UTC Physics 183 – Simple Pendulum
THE SIMPLE PENDULUM
Objective:
To investigate the relationship between the length of a simple pendulum and the period of
its motion.
Apparatus:
String, pendulum bob, stopwatch, meter stick, logarithmic graph paper, a computer with
ULI interface, and a photogate.
Theory:
A simple pendulum consists of a small bob of mass m
suspended by a light (massless) string of length L fixed at its
upper end. When released from point P, the mass swings
through point O and to a point of equal height to point P, and
back to point P over the same path (Figure 1).
The force that restores the pendulum to its
equilibrium position, point O, after it is disturbed is the
component of the weight of the bob, mg, that is tangential to
the circular path, denoted as F
t
:
F
t
= - m g sin
θ
(eq. 1),
where the negative sign denotes acceleration due to gravity is
in the downward direction. For small angles, the value for the
angle
θ
(measured in radians) is equal to the sine of that angle, or sin
θ
=
θ
. Substituting into equation 1
yields F
t
= - m g
θ
.
Furthermore, the arc length that the bob swings through in traveling between
points O and P is denoted as S and is related to the length of the string L (or radius of circular motion)
by
θ
= S / L. Thus,
S
L
mg
F
t
=
(eq. 2).
For small angles, the arc length is adequately approximated by the straight line distance between points
O and P.
The period, T, of an object in simple harmonic motion is defined as the time for one complete
cycle. For the simple pendulum, this is specifically given by:
g
L
T
π
2
=
or
2
1
2
L
g
T
=
(eq. 3),
where g is the acceleration due to gravity, 9.8 m/s
2
. Equation 3 indicates that the period and length of
the pendulum are directly proportional; that is, as the length, L, of a pendulum is increased, so will its
period, T, increase. However, it is not a linear relationship. The period increases as the square root of
the length. Thus, if the length of a pendulum is increased by a factor of 4, the period is only doubled.
This is a logarithmic relationship. A more general form of equation 3 is:
T
=
k
L
n
(eq. 4),
1