Exploration Guide: Simple Harmonic Motion
When an object moves back and forth in a regular pattern, or oscillates, its motion is said to be harmonic.
Springs and pendulums are two types of objects that undergo simple harmonic motion. Because they oscillate
with a constant period, they can be used to measure time. If you have seen a grandfather clock, you have seen a
pendulum used to measure time. Springs can be found in old-fashioned pocket watches and wrist watches.
One of the great challenges of the 18 t h century was to design and build a clock that would keep accurate time
while at sea. An accurate clock is a critical component in the process of finding longitude at sea. Because the
rocking of the ship disrupted the action of all pendulums, a new kind of timepiece that used springs would have to
be used instead. The story of the development of the first navigational clocks is a fascinating one, masterfully told
in the book Longitude, by Dava Sobel.
Measuring Periods
In this activity, you will measure the period of a pendulum and a spring. The period is the time it takes an object in
simple harmonic motion to complete one full cycle. For a spring, this is the time required for the spring to move up
and down. For a pendulum, the period is the time required to move back and forth once.
1.
Observe the movement of the spring. Notice that in each oscillation, the blue mass on the end of
the spring returns to the same low point. Click POINTER at the bottom of the Gizmo™, and drag a
pointer so that it just touches the mass at the low point of its movement.
a.
When the mass on the spring touches the pointer, press the green button on the left
side of the Stopwatch to start the clock. Count ten full cycles of movement for the
spring, and when the mass at the end of the spring touches the arrow at the end of the
tenth time, click the green button again.
b.
c.
Divide the total time by ten to find the measured period of spring system.
Reset the timer by clicking the red button on the right side of the Stopwatch.
Repeat your measurement of the spring period to assess the accuracy of your method.
How close was this measured period to the original measured period?
d.
Experiment with the mass (m) and spring constant (k) sliders. (The spring
constant measures the stiffness of the spring.) How does each variable affect the
period of the spring?
2.
Now, you will measure the period of the pendulum. This simple pendulum is constructed of a
weight, or bob, attached to a weightless string. Drag a pointer so that it just touches the mass at
the left-most point of its swing.
a.
Reset the timer. When the pendulum bob touches the pointer, start the timer. At
the end of the tenth swing, stop the timer.
b.
Calculate the period of the pendulum. Is its period longer or shorter than the period
of the spring that you measured?
c.
Experiment with the mass (m) and length (L) sliders. How does each variable affect
the period of the spring?
Factors Affecting Periods
In this activity, you will perform several experiments to determine the effect of each variable on the spring and
pendulum.
1.
Check that the k and g sliders below the spring are set to their default values, k = 100.0 N/m ,
and g = 9. 8 m/s 2 . (The value g = 9.8 m/s 2 is the acceleration due to gravity on Earth.) Set m,
the mass on the spring, to 0.5 kg.
a.
What do you notice about the spring? Does it reach as low as before? Explain why
you think this is.
b.
To make your measurements, set the Sim. speed slider to a lower setting. Measure
the period of the spring using the method you practiced in the first activity. What is the
period for an oscillation of this system? Record this value in your notes.
c.
Increase m to 2.0 kg, and leave the other values unchanged. How does this change
the motion of the spring?
d.
What is the period of the spring now? How does the period compare to the period
when the mass was 0.5 kg? (What value was the period multiplied or divided by?)
What was the value of m multiplied by?
e.
You should have found that, when the mass was multiplied by 4, the period was
multiplied by 2. Now, measure the period of the system when the mass is 0.2 kg and
1.8 kg. (Note: for small masses, you may need to slow the simulation speed using the
Sim. speed slider.) Record your results in a data table. When mass is multiplied by 9,
what was the period multiplied by?
f.
Set the Sim. speed to its slowest setting, and measure the period when m is 0.1
kg. Based on the patterns you have seen so far, predict the period when the mass is
multiplied by 16 (m = 1. 6 kg ). Use the Gizmo to check your answer.
g.
Based on what you have seen, write a general rule or expression to summarize how
changing the mass affects the period of the spring. (Hint: Use the word square root in
your rule.)
2.
Check that the mass of the pendulum bob is 1.0 kg and the length of the pendulum string is 2.0 m.
Record the period of the pendulum in this situation.
a.
What do you predict will happen to the period of the pendulum if you increase the
mass of the pendulum bob?
b.
Increase the mass of the bob to 2.0 kg and measure the period. Was your
prediction correct?
c.
Try another value for the mass of the pendulum bob. Does the period of the
pendulum ever change?
3.
The mass of a pendulum bob does not affect the period of the pendulum. What about the length of
the string? To test this, decrease the length of the string to 0.5 m and measure the period.
a.
Compare the 0.5 m period to the 2.0 m period you measured in the previous step.
What happens to the period when the length of the string is multiplied by four?
b.
Now, measure the periods of the pendulum for string lengths of 0.2 m and 1.8 m.
What happens to the period when the length is multiplied by nine?
c.
Try lengths of 0.1 m and 1.6 m. How does the period change when length is
multiplied by 16? Based on your observations, write a general rule or expression for
the relationship between string length and period. How is this relationship similar to the
relationship between the mass and period of a spring?
4.
Now, using a similar method with the spring system, see how changing the spring constant, k,
affects the period of the spring's oscillation. Keeping g and m constant, plan an experiment where
you change the spring constant by multiples of 2, 3, and 4. From your observations, write a general
rule for the relationship between the spring constant and period. How is this relationship similar to
the other relationships you have discovered?
5.
Changing the acceleration due to gravity will affect the period of one of the two systems. Plan an
experiment to see which system is affected by the change.
a.
What variable will you change in the experiment? The variable that you change in
an experiment is called the independent variable. This will cause changes in other
variables, which you will need to measure. These other variables are dependent
variables. What are the dependent variables in this experiment?
b.
c.
Which system was affected by the change in gravity? Which system was unaffected?
Based on your observations, plan another experiment to determine the relationship
between the acceleration due to gravity and the period of the system it affects. If
possible, compare your results and discuss your findings with your classmates and
teacher.
Assessment Questions
1.
1. Suppose that it takes 0.6 seconds for a mass on a spring to move from its highest position to its
lowest position. What is the period of the spring?
A. 0.3 seconds
B. 0.6 seconds
C. 1.2 seconds
D. 2.4 seconds
2.
2. Which of the following factors has no effect on the period of a pendulum?
A. mass of the pendulum bob (m)
B. length of the pendulum's string (L)
C. acceleration due to gravity (g)
D. all of these factors affect the period of a pendulum
3. The period of a spring system is 1.4 seconds. The spring is moved to a place where the
acceleration due to gravity is
two times as large as in the first experiment. What is the new period of the spring system?
A. 0.7 seconds
B. 1.4 seconds
C. 2.0 seconds
D. 2.8 seconds
4. The period of a pendulum is 1.4 seconds. The length of the pendulum is increased by a factor of
4. What is the new
period of the pendulum?
A. 0.7 seconds
B. 1.4 seconds
C. 2.8 seconds
D. 5.6 seconds
5. A mass-spring system oscillates with a period of 2.2 s. What would be the period of the system if
the mass and the
spring constant were both increased by a factor of four?
A. the period would decrease by half
B. the period would remain the same
C. the period would be multiplied by the square root of two
D. the period would double