Simple Harmonic Motion
Factors that influence the change in the period of a SHM
Victor Jeung, Terry Tong, Cathy Liu, Jason Feng
November 25th, 2011
1
Simple Harmonic Motion
Abstract
Simple harmonic motion accurately models the motion that a mass or a pendulum exhibits during
movement either from their equilibrium point on a spring to the stretched distance, or when a
pendulum swings from side to side. A simple harmonic motion will remain in motion as long as the
system does not experience any type of external force such as friction, or an opposite applied force. Our
experiment will demonstrate how a pendulum and a mass spring system behave in a simple harmonic
motion.
Introduction
Hypothesis
The pendulum and the spring mass system will behave in a simple harmonic motion and are dependent
on 2 variables that are different to each system. For the mass spring system, the period of the motion
will depend on the mass that is hanging and the spring coefficient. The pendulum will depend on the
length of the string.
Theory
Variables
Variables are separated into two categories. The mass-spring system and the pendulum each have only
two different variables. The period of the mass-spring system is dependent upon the spring coefficient
and the mass of the hanging weight, while the pendulum system is dependent upon the length of the
spring and the mass that is being swung.
Equation
⃗
√
√
(In simple spring system)
√
√
(In simple spring system)
⃗⃗⃗⃗
T is the period
L is the length of the pendulum
f is the frequency
g is the acceleration due to the gravity
m is the mass of the object
F is the force due to the spring
k is the coefficient of the spring
X is the displacement from the object to the equilibrium point
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Simple Harmonic Motion
Experiment
Conducting the experiment
Our experiment for this lab is comprised of using a retort stand and either a string or a spring to hold the
hanging mass.
Apparatus





A set of masses
Timers
String
2 Different Types of Spring
Meter stick
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Simple Harmonic Motion
4
Simple Harmonic Motion
5
Simple Harmonic Motion
Conducting the Experiment
Mass-Spring
Conducting the mass-spring test was done by first attaching a weight on to the end of a spring and
letting it settle to an equilibrium state (in which the mass is not in an SHM). Then stretching it to certain
amplitude and letting it go in to a SHM. When the SHM begins, the timers start counting and the
counters count how many times the mass has gone through its equilibrium point and back until 10
which we stop and record the time.
Pendulum
The pendulum experiment was conducted by first attaching a mass to a measured string. The mass is
then let go at a that we measured to equal 60, and let go to swing back and forth. The timer and
counter would both start when the mass is let go at , and would stop after the pendulum has gone
through 10 periods of SHM.
Data
length(m)
mass(kg)
0.1
0.2
0.3
0.1
0.4
0.5
0.02
0.05
0.3
0.10
0.15
0.2
trial
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
Pendulum
time of 10 swings (s)
8.6
8.6
10.9
11.0
12.7
12.5
14.3
14.4
16.1
16.1
12.6
12.3
12.6
12.5
12.6
12.8
13.1
12.8
13.0
13.1
period (s)
0.86
0.86
1.09
1.10
1.27
1.25
1.43
1.44
1.61
1.61
1.26
1.23
1.26
1.25
1.26
1.28
1.31
1.28
1.30
1.31
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Simple Harmonic Motion
K type
mass (kg)
0.05
K1
0.1
0.2
0.05
K2
(K1> K2)
0.1
0.2
trial
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
Spring
time for 10 up and down (s)
4.3
4.4
4.3
4.2
4.4
5.9
6.1
6.1
5.8
6.0
7.9
7.9
8.0
7.9
7.7
7.0
6.9
6.9
7.1
6.8
9.8
10.1
9.9
9.9
10.0
15.1
14.7
14.5
15.0
14.3
period(s)
0.59
0.61
0.61
0.58
0.60
0.43
0.44
0.43
0.42
0.44
0.79
0.79
0.80
0.79
0.77
0.70
0.69
0.69
0.71
0.68
0.98
0.10
0.99
0.99
0.10
0.15
0.15
0.15
0.15
0.14
amplitude(m)
0.005
Data analysis
The data we collected show the relationship between different variables in both pendulum lab and
spring lab. In a simple pendulum system, only the change of the length affects the period, but not the
change of mass. In a simple spring system, both mass and K value affect the period.
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Simple Harmonic Motion
Conclusion
In conclusion, we proved that our hypothesis is right. The period of an object in pendulum-mass system
depends on the length of the string only, which has a direct ratio with the period. For the spring-mass
system, the period depends on both the coefficient of spring and the mass of the hanging object. The
spring constant has an inverse ratio with the period and the mass has an direct one.
There were some random, systematic and human errors in our lab. These were:

The random error of the spring moving left and right while in motion. This caused the spring not
to reach the full amplitude, and caused a shortening of the period as well as an increased
frequency.
 The human error of reaction time when counting the number of cycles completed. Because the
spring and pendulum were moving quite fast, it was hard to count the number of cycles that had
completed.
 The systematic error of not using ideal equipment such as a frictionless and massless string for
the pendulum and an ideal spring for the mass-spring system.
 The systematic error of air resistance (friction) for the simple pendulum and the mass-spring
system. Due to air resistance (friction), the pendulum and mass-spring system started to slow
down and not return to the same height as more cycles were completed.
 The random error of not releasing the pendulum and mass-spring system at the same height for
each trial. Releasing the pendulum and spring at slightly different heights resulted in differences
of potential energy during each trial.
 The random error of reading the location of the mass in the mass-spring system while it was in
motion. It was hard to determine where the mass was at since the mass-spring system was
moving fast.
If we were to complete the simple harmonic motion lab again we would:





Tape a ruler onto the retort stand to make it easier to track the location of the mass and
pendulum while in motion.
Ensure that the pendulum and spring are released at the same height for each trial by marking
off where the pendulum and spring are released.
Use a less stiff spring so that the motion of the mass-spring system is slower making it easier to
count the number of cycles completed and the location of the spring while in motion.
Try using different springs with different k values to determine how the stiffness of the spring
affects the period and frequency of the mass-spring system.
Try using different lengths of string to determine the relationship between the length of the
string and period as well as frequency.
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Simple Harmonic Motion