The Simple Pendulum

Recall from lecture that a pendulum
will execute simple harmonic motion
for small amplitude vibrations.

Period (T) - time to make one
oscillation

Frequency (f) - number of
oscillations per unit time
Period
Frequency
1 vib r ation/ sec
1 sec/ vib r at ion
2 v ib r ations / sec
1
2
sec/ v ib r at ion
3 v ib r ations / sec
1
3
sec/ v ib r at ion
4 v ib r ations / sec
1
4
sec/ v ib r at ion
1
2
vib r ation/ sec
2 sec/ v ib r at ion
Period 
1
Frequency
In symbolic form
or
1
T
f

The period is independent of the mass of the
pendulum.

The period depends on the length of pendulum.

It also depends on the amplitude (angle of swing).

If the displacement angle is small (less than 100),

then the period of the pendulum depends primarily
on the length (l ) and the acceleration due to
gravity (g) as follows.
T
l
 2
g
It must be emphasized again that this equation is good
for small angles of vibration but not for large.

Squaring both sides of the equation
yields
T

2
l
 4
g
2
Let’s rewrite this equation to get
T
2
4 π 2
  g


l


2 

2
4
π
T   g l



This is of the form (from last week’s lab)
y m x b
T 2 is y
4 2/g is m
l is x
and b will equal zero

Therefore by plotting T 2 versus l and
using the slope of this curve one can
determine the acceleration due to
gravity g. The slope is
slope

4π
g
2
Multiply both sides
of the equation by g
and get
(g) slope

This reduces to
2
(g)
slope

4π
Now divide both
slope  g
slope
sides by the slope
(g)

to get
slope
which reduces to
g
4π
(g)
g
2

4π
2
2
π
4
slope
2
4π
slope
Variation of g
Around the World
Below are listed factors that affect the
local value of the acceleration due to
gravity.
 Altitude
 Latitude
 Geology
T
l
 2
g
As a review, note that the period T increases if l increases
or if g decreases. T will decrease if l decreases
or if g increases. (It is important that you know this for
both lecture and laboratory exams.)
Purpose of Today’s Experiment
You will determine the local value of the
acceleration due to gravity by studying
the motion of a simple pendulum.
Note: Pendulums are used in a variety of
applications from timing devices like
clocks and metronomes to oil prospecting
devices.