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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.

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