# Lesson - Simple Harmonic Motion Pendulum

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```					Lesson - Simple Harmonic Motion: Pendulum
Name:__________________________
The applet simulates the simple harmonic motion of a pendulum.

Prerequisites

Students should be familiar with the definitions for period and frequency.

Learning Outcomes

Students will develop an understanding of simple harmonic motion as it applies to a pendulum. The will
also be able to apply the equation describing the period of a pendulum and describe the sinusoidal
nature of simple harmonic motion.

Instructions

Students should know how the applet functions, as described in Help and ShowMe. Many of the step-by-
step instructions in the following text are to be done in the applet.

Contents

Observing Simple Harmonic Motion
Equation for the Period of a Pendulum
Simple Harmonic Motion is Sinusoidal

Observing Simple Harmonic Motion
Hundreds of years ago, Galileo Galilei watched a candle chandelier
swinging back and forth on a long chain in a cathedral. Galileo wondered
what determined the period of the pendulum. What is the effect of
changing the length of the chain? If the mass of the chandelier was
changed, would that effect the period of vibration? And, if the amplitude
(size of swing) was increased or decreased, would the period be effected?
To answer these questions he did an experiment similar to the one you are

At rest, the pendulum hangs at the equilibrium position. The pendulum is
given potential energy when the bob is pulled to one side. When it is
released the pendulum is pulled by gravity toward the equilibrium position.
The potential energy is converted into kinetic energy until, at the
equilibrium position, the pendulum reaches its maximum speed. The                   Figure 1
pendulum then continues through the equilibrium position to end up

Lesson – Simple Harmonic Motion – Pendulum                                                   1 of 8
instantaneously at rest on the opposite side from where it started. The kinetic energy has been
converted back into potential energy. Then gravity pulls the pendulum back through the equilibrium
position until it reaches its starting position again. One cycle or vibration has been completed.

A moving pendulum exhibits simple harmonic motion. This means that the pendulum is vibrating with a
constant frequency or period of motion and that there is a restoring force directed towards the a central
equilibrium point. Furthermore, the magnitude of the restoring force is proportional to the displacement
from the equilibrium point. The applet will be used to show the direction and magnitude of the restoring
force.

On the applet, select the acceleration vector for display (                   ) and click
the vector components button ( ). Press play and observe the motion of the pendulum and the
corresponding acceleration vector. The acceleration vector is proportional in magnitude to the restoring
force by Newton's Second Law (F = ma). Based on your observations, draw the restoring force on each
of the images below. The first one has been completed as an example.

Equation for the Period of a Pendulum

The period of a pendulum is based on several known parameters. The applet will be used to identify
which parameters contribute to the period of a pendulum.

On the applet, determine the period of the pendulum by doing the following:

   reset the applet ( )
   turn on the data display ( )
   press play and carefully watch the pendulum move through 10 complete cycles (one cycle is
complete every time the pendulum returns to the starting position)
   stop the applet at the end of the tenth swing
   record the time for ten cycles from the data display

time for ten cycles: _______________

divide the recorded time by ten to calculate the time of one cycle:_______________________

Lesson – Simple Harmonic Motion – Pendulum                                                       2 of 8
The time required to complete one cycle is the period of the pendulum. Now that we know this, we will
systematically investigate one parameter at a time by answering the following four questions.

Question 1: Does the angle of release effect the period?

On the applet, determine the period of the pendulum at a different angle of release by
doing the following:

   reset the applet ( )
   turn on the data display (   )

   change the angle of release to the maximum value of 0.52 (             )
   press play and observe 10 complete cycles

time for ten cycles: _______________

divide the recorded time by ten to calculate the time of one cycle:_______________________

Has the period changed as a result of changing the angle of release?

Question 2: Does the mass of the pendulum effect the period?

On the applet, determine if the period of the pendulum is effected by the mass of the
pendulum by doing the following:

   reset the applet ( )
   turn on the data display (   )

   change the mass of the pendulum to the maximum value of 1.00 kg (               )
   press play and observe10 complete cycles

time for ten cycles: _______________

divide the recorded time by ten to calculate the time of one cycle:_______________________

Has the period changed as a result of the change in the mass of the pendulum?

Lesson – Simple Harmonic Motion – Pendulum                                                      3 of 8
Question 3: Does the length of the pendulum effect the period?

On the applet, determine if the period of the pendulum is effected by the length of the
pendulum by doing the following:

   reset the applet ( )
   turn on the data display (   )

   change the length of the pendulum to the maximum value of 2.0 m (               )
   press play and observe 10 complete cycles

time for ten cycles: _______________

divide the recorded time by ten to calculate the time of one cycle:_______________________

Has the period changed as a result of the change in the length of the pendulum?

Question 4: Does the acceleration due to gravity effect the period?

On the applet, determine if the period of the pendulum is effect by the acceleration due
to gravity by doing the following:

   reset the applet ( )
   turn on the data display (   )

   change the acceleration due to gravity to the maximum value of 20.0 m/s 2 (             )
   press play and carefully watch the pendulum move through 10 complete cycles

time for ten cycles: _______________

divide the recorded time by ten to calculate the time of one cycle:_______________________

Has the period changed as a result of the change in the acceleration due to gravity?

Summarize your findings from Exercise 2-6 by listing the parameters that do effect the
period of the pendulum and the ones that do not have an effect.

Lesson – Simple Harmonic Motion – Pendulum                                                         4 of 8
The Period of a Pendulum
The period of pendulum is the time required to complete one cycle or swing. It is defined by only the
length of the pendulum and the acceleration of gravity.

Expressed as an equation:

(1)

Quantity                               Symbol           SI Unit
period                                    T                s
length                                     l               m
acceleration due to gravity               g              m/s2

The period depends on the length of the pendulum and the acceleration of gravity. For a complete
derivation of this equation see pages 350 to 355 in the Fundamentals of Physics.

The pendulum had immediate application in several areas of science. In addition to keeping good time,
simple pendulums were used to measure the strength of the Earth's gravitational field at various
locations on the surface of the Earth. Variations in the acceleration readings can be used to indicate the
possible presence of heavy ores under the ground. In a sense, a precise pendulum is a metal detector!

Example Problem 1: If a pendulum is 80.0 cm long, what is its
period and frequency of vibration?

Lesson – Simple Harmonic Motion – Pendulum                                                      5 of 8
Example Problem 2: On an unexplored planet, a probe found
that a 50.0 cm pendulum completed 20 swings in 33.6 s. What
is the acceleration of gravity on this planet?

Calculate the period of a 1.50 m pendulum. Verify your answer using the applet.

On the planet Xeon, a pendulum having a length of 95.0 cm swings with a frequency of
1.50 Hz. What is the acceleration due to gravity on Xeon?

A pendulum in a grandfather clock completes 60 cycles every minute. Calculate the
period and frequency of the motion. How long must the pendulum be in order to maintain this period?

Lesson – Simple Harmonic Motion – Pendulum                                                  6 of 8
Simple Harmonic Motion is Sinusoidal

Simple harmonic motion is sinusoidal in nature.
Tracking the velocity of the pendulum as it
rotates through several cycles produces a
curve that has precisely the same shape as a
sine curve, so called because it is like the graph
of the sine function in trigonometry. Figure 2
illustrates the sinusoidal nature of the
pendulum's velocity.

Figure 2

Simple harmonic motion (SHM) is also based on uniform circular motion (UCM). Note the x-axis on
Figure 2 shows radian units. Why is this? The circumference of a circle is equal to 2 radians and for
each cycle of the pendulum it completes one circle. Other similarities between SHM and UCM can be
found in the equations.

Manipulating Equation (1) above, in terms of acceleration, gives:

which is suspiciously similar to                from uniform circular motion.

This is not surprising if we recall that UCM is vibratory with a constant period and frequency. Since both
equations describe periodic, vibratory motion, they should have the same form.

Observe uniform circular motion and simple harmonic motion at the same time by doing
the following:

   reset the applet (   )
   display the reference circle (   ), vector components (   ) and extrapolation lines (   )
   select the velocity vector for display (                  )
   press play

Describe one similarity and one difference between the velocity vector on the reference circle and the
velocity vector on the pendulum.

Lesson – Simple Harmonic Motion – Pendulum                                                         7 of 8
Without changing any of the setting from Exercise 11, turn on the graphing function (    )
and select the velocity vector as illustrated in Figure 3.

Figure 3

a. How many rotations does the reference circle make for every complete wave (              ) drawn
on the graph?

b. How many complete swing cycles does this represent on the pendulum?

Lesson – Simple Harmonic Motion – Pendulum                                                      8 of 8

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