A pendulum consists of a weight
suspended on a rod, string or wire. When the weight or
bob is moved and let go, the pendulum will swing back
and forth in a regular periodic motion. The affect of
gravity on the bob results in the periodic motion and
its length determines the frequency of its swing.
Pendulums have been used in clocks for hundreds of
years, because the motion is so regular.
Questions you may have include:
-
What are some properties of a
pendulum?
-
How do your determine its frequency?
-
What are some applications of a
pendulum?
This lesson will answer those questions.
Simple pendulum A
simple pendulum consists of a rod or wire attached at a
pivot point. On the other end of the rod is a weight or
bob. When pulled to the side and let go, the bob will
swing back and forth due to the affect of gravity.
The drawing below shows the different factors involved
in a pendulum.
Simple pendulum A pendulum has some
interesting properties, concerning its frequency (how
many times it goes back and forth per second).
Dependent of length The frequency
of the pendulum is dependent on the length (L) of the
string or wire. The shorter the wire, the greater the
frequency or how fast it goes back and forth.
Independent of amplitude The
frequency is independent of the amplitude (A) of the
swing, provided the initial angle (a) is not large. At
larger angles, there is a slight change in the
frequency. Independent of mass
Also, the frequency is independent of the mass of the
bob. In other words a pendulum with a heavy bob will
move at the same rate as one with a lighter weight bob.
But this only makes sense, since the acceleration of
gravity on a falling object is independent of the mass
of the object.
When a pendulum moves, there is some air resistance on
the bob and rod or wire. There is also friction at the
pivot point. These resistive forces reduce the amplitude
of the swing, such that after a while the pendulum will
come to a stop. These forces are called damping forces.
In the demonstration above, you may note that the
amplitude of the swing gets smaller with time. There is
a slight damping factor included in the simulation.
Calculations Knowing the length of the
pendulum, you can determine its frequency. Or, if you
want a specific frequency, you can determine the
necessary length.
The equation to calculate the frequency of a pendulum
is: f = (1/2pi) * sqrt(g/L)
or f = (1/2π) * √(g/L)
where
-
f = frequency in cycles per second
(Hertz or Hz)
-
pi = 3.14 (it is also written as the
Greek letter π, but that may not show up in some
older browsers)
-
sqrt means the square root of what
is included in the parentheses (sqrt is also seen as
the symbol √, but that may not show up in some older
browsers)
-
g is the acceleration of gravity
(9.8 m/s2 or 32 ft/s2)
-
L is the length of the rod or wire
in meters or feet
If you wanted to
find the length for a given frequency, you can write
that equation as:
L = g/(4*pi2*f2)
or L = g/(4*π2*f2) For
example, the length of a pendulum that would have a
frequency of 1Hz (1 cycle per second) is about 0.25
meters. Two
problems, applying those equations are:
-
What is the frequency of a 5
kilogram pendulum on a 2 meter wire?
-
What is the length of a clock
pendulum with a frequency of 1Hz in feet?
Applications The
two major applications of pendulums are in telling time
and the Foucault Pendulum.
The most common application of the pendulum is to use
its regular motion to control the motion of the hands of
a clock. This is still seen in the older grandfather
clocks. Every time the pendulum goes back and forth, it
moves a gear one notch. Gears are then used to move the
hands of the clock. The length of the
pendulum can be adjusted slightly, if the clock is
running too fast or slow. Another interesting application
is called the Foucault Pendulum. This pendulum will
demonstrate the Earth's rotation. The is
a Foucault Pendulum large pendulum that is often several
stories high. The reason it is so large is so that it
will keep swinging over a longer period of time.
Friction forces using damp a smaller pendulum and cause
to finally stop after a relatively short time. In 1848, Jean Foucault
discovered that when a large pendulum swings over a long
period of time, the pendulum appears to be changing
directions during the day. What is really happening is
that the pendulum is moving in the same direction, but
the Earth has rotated under the pendulum.
Although there are now Foucault Pendulum's in many
locations, the most famous Foucault Pendulum is at the
Pantheon in Paris, France. The picture below shows the
size of the pendulum and the scale at the bottom to
indicate the positions at different times of the day.
Foucault Pendulum in Paris To explain how the Foucault
Pendulum works, consider putting a pendulum exactly at
the North Pole or South Pole. While the Earth rotated on
its axis, the pendulum would continue to swing in the
same direction in space. It would appear as if the
pendulum was slowly changing directions, but in reality
it is the Earth that is revolving underneath the
pendulum. This same phenomenon will
happen at locations other than the poles, except that
the reason is not as obvious. In
conclusion A pendulum is a suspended
weight that swings back and forth in a regular periodic
motion. The length of the pendulum determines its
frequency, while the weight of the bob does not affect
the frequency. Pendulums have been used in clocks for
hundreds of years, because the motion is so regular. |