Length of Pendulum and Period
Date: April 28, 2011
Why does length of a string affect the period of a pendulum?
The motion of a pendulum can be explained by equations describing
rotation. The component of the gravitation force driving a free
swinging pendulum depends on the angle it makes with the
vertical. That angle is a function of the length of the
pendulum. In addition, the resistance to the swinging is dependent
on the rotational moment of inertia which also depends on the length
of the pendulum. These two factors involving length do not cancel
each other out. For a simple pendulum the moment of inertia is mass
x Length squared. The result is
Period of Pendulum = 2Pi x (square root of L/g) where L is length
of simple pendulum and g is the gravitation constant.
David S. Kupperman
Consider two pendulums of different lengths, both moving at the same
speed when at the bottom of the swing. Each will swing up to the same
height, the same potential energy. The very short pendulum moves upward
quickly, over a very short distance. If the pendulum is too short, it
may even loop over the top. The very long pendulum will have to move
over a long distance before rising up much. Because both pendulums
would have the same speed at the bottom, the long pendulum would take
more time to get to the maximum height. This corresponds to a longer
Dr. Ken Mellendorf
Illinois Central College
Well, if you have two pendulums, both pulled back to the 30-degree position and let go,
the one with the longer string has a longer distance sideways to go, before it reaches center.
Or suppose you pulled each pendulum back a fixed sideways distance before letting go.
Then the one with the longer string would be travelling down a shallower slope,
so it would not accelerate towards the center as fast.
Draw two circles on the blackboard, one big and one smaller.
The point in the center is the attachment point of the string,
and the circle is the path the weight must travel.
Draw two vertical lines across each of the circles, offset to left and right of center
by about half the radius of the smaller circle.
It will be apparent to the eye that the slope at the intersection points
is shallower for the larger circle, and steeper for the smaller circle.
I think they will understand that the steeper slope makes for faster acceleration sideways.
It is quite a bit like a ball falling off a table or rolling down a ramp.
If the table is 4 times as high, it takes 2 times as long for the ball to hit the floor.
The rule with accelerating-motion such as gravity is:
time goes as square root of the distance,
or specifically: distance = 1/2 * acceleration * time^2
I am not sure how familiar your students are with square-roots, squares, and functions in general.
But square-root seems to be the usual compromise response
between being unchanging and being proportional.
I think if you had a two-foot pendulum and a 6-inch pendulum, (or 1-foot and 4-feet),
then it would be very clear to the eye
that the short one did two swings for every one swing of the long one.
The longer the distance between mass (pendulum) and pivot, the
greater the "rotational inertia". In real words, it means the
longer the pendulum the harder it is to move. It looks "heavier"
than the same mass on a shorter pendulum. As a result it moves
slower with the same gravity. After all we ca get a light car up to
running speed faster than a heavy car or bus.
Hope this helps without getting too technical. A good high school
physics text will go into more detail and math should you want more detail.
Take care and good luck from a former 8th grade science teacher!
R. W. "Bob" Avakian
Arts and Sciences/CRC
Oklahoma State Univ. Inst. of Technology
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Update: June 2012