Mass Independence of Pendulum Period
How come changing the weight of a pendulum does not affect the
number of times it swings in 20 seconds?
It is because a larger pendulum mass requires a greater force to move (or
accelerate) it. The amount of force is proportional to the mass. In other
words, if the mass is twice a big, it requires twice the force to accelerate
it at a given rate... but the force is just that... twice as great. (Gravity
pulls harder on a larger mass.)
It will always move from the top of the swing to the low point in the same
A pendulum is like a falling body. Galileo showed that the speed of
heavy objects is the same as light ones (except for light ones that are
slowed by air resistance, which is not the case for simple pendulums
with reasonably heavy bobs). Since the motion of a pendulum is
essentially an object falling toward the center of motion and masses
fall at the same rate, the swing time of the pendulum does not depend on
Nevertheless there is one very interesting problem associated with a
pendulum. In order for the swing time to be independent of mass the
ratio of the inertial mass (the measure of resistance that the bob has
to acceleration) to the gravitational mass (the measure of the
gravitational force) has to be the same for all materials. So far no
one has been able to show that there is a change of ratio of inertial to
gravitational mass if one changes the bob material. If there is a change
it would have to be less than one part in 100 billion. This observation
was important in the development of Einstein's general theory of
David S. Kupperman
The driving force for a pendulum is gravity. If the pendulum has twice
the mass, gravity pulls twice as hard. Mass is also how hard an object
resists the forces it feels. It is much more difficult to start motion
for a bowling ball than for a ping-pong ball.
Now put these together. A pendulum with twice the mass feels twice the
pull, but also has twice the resistance to that pull. These two effects
balance out. A pendulum with twice the mass still experiences the same
effect. The mass of a pendulum does not affect how it moves.
Dr. Ken Mellendorf
Illinois Central College
There is a story that Galileo Galilei sat in Church at the Cathedral of Pisa
asking himself the same question.
According to one story he noticed that the chandelier in the cathedral was
swinging, but what attracted his attention was that it was swinging in time
to the music that was being played. Since the music was played to a strict
rhythm, was the chandelier swinging to a strict rhythm?
From about 1602, he began serious study of the properties of pendula (plural
of pendulum). His studies showed that the period of the pendulum (that is :
the time it takes to swing from left, to right and back to its start position )
was determined by the length of the pendulum, and not by the mass (or weight) of
the lump at the end (which is usually called a bob) He also proposed that the
period of the pendulum was not affected by the amplitude (size) of the swing,
although it has since been shown that this is true for smaller amplitudes, but not
for very large ones.
In other words, once the length of the string was set, the timing of the pendulum
was set no matter how much the bob weighed, or (within reasonable limits) how far
the pendulum was set swinging at the start, or how little it moved at the end.
This was very useful as a timing device, and has been used in clocks of various
sorts since soon after Galileo's discoveries.
Curiously, while the mass of the bob has no effect, it turns out that the force
of gravity does. On the moon, not only would the bob weigh less, but it would also
swing more slowly.
Now, to finally answer your question WHY? - the formula for calculating the period
of the pendulum is
T = 2 pi * Square root of l/g where pi is the constant 3.14159.... G is
gravity and l is the length of the pendulum. You will notice that mass is
not included in the equation, because it has no effect. The force required to
MOVE the mass - (its momentum) and the force provided by gravity (its weight)
are both determined my the mass of the object, so they cancel each other out and
that is why there is no effect.
It is because the force accelerating the pendulum comes from its weight. The more
massive the pendulum is, the greater its weight, in strict proportion. The inertia,
that is, the resistance of the pendulum to acceleration by a force, is also in
strict proportion to its mass. Since both the force and inertia vary in strict
proportion to the mass, the two effects cancel out and the acceleration is
independent of mass.
In other words:
The more massive something is, the more force is required to make it accelerate
at a given rate.
The more massive something is, the more force gravity exerts on it.
The net result is that acceleration from the force of gravity dies not change as
an object's mass changes!
Richard Barrans, Ph.D., M.Ed.
Department of Physics and Astronomy
University of Wyoming
Click here to return to the Physics Archives
Update: June 2012