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• 1 Simple Harmonic Motion

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

• 2 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

f is the frequency

m is the mass of the object

k is the coefficient of the spring

L is the length of the pendulum

g is the acceleration due to the gravity

F is the force due to the spring

X is the displacement from the object to the equilibrium point

• 3 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

• 4 Simple Harmonic Motion

• 5 Simple Harmonic Motion

• 6 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

Pendulum

length(m) mass(kg) trial time of 10 swings (s) period (s)

0.1

0.1

1 8.6 0.86

2 8.6 0.86

0.2 1 10.9 1.09

2 11.0 1.10

0.3 1 12.7 1.27

2 12.5 1.25

0.4 1 14.3 1.43

2 14.4 1.44

0.5 1 16.1 1.61

2 16.1 1.61

0.3

0.02 1 12.6 1.26

2 12.3 1.23

0.05 1 12.6 1.26

2 12.5 1.25

0.10 1 12.6 1.26

2 12.8 1.28

0.15 1 13.1 1.31

2 12.8 1.28

0.2 1 13.0 1.30

2 13.1 1.31

• 7 Simple Harmonic Motion

Spring

K type mass (kg) trial time for 10 up and down (s) period(s) amplitude(m)

K1

0.05

1 4.3 0.59

0.005

2 4.4 0.61

3 4.3 0.61

4 4.2 0.58

5 4.4 0.60

0.1

1 5.9 0.43

2 6.1 0.44

3 6.1 0.43

4 5.8 0.42

5 6.0 0.44

0.2

1 7.9 0.79

2 7.9 0.79

3 8.0 0.80

4 7.9 0.79

5 7.7 0.77

K2 (K1> K2)

0.05

1 7.0 0.70

2 6.9 0.69

3 6.9 0.69

4 7.1 0.71

5 6.8 0.68

0.1

1 9.8 0.98

2 10.1 0.10

3 9.9 0.99

4 9.9 0.99

5 10.0 0.10

0.2

1 15.1 0.15

2 14.7 0.15

3 14.5 0.15

4 15.0 0.15

5 14.3 0.14

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.

• 8 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.