Lab 07: Simple Harmonic Motion

Lab 07: Simple Harmonic Motion

Learning Objectives

Define simple harmonic motion, period, frequency, amplitude, and restoring force.
Describe the relationship between period and amplitude in a simple pendulum.
Describe the relationship between period and length in a simple pendulum.
Describe the relationship between period and mass in a simple pendulum.
Describe the relationship between period and mass in a vertical spring system.
Use measured time for 10 vibrations to compute period and frequency.
Construct and interpret a graph of mass versus period squared for a spring system.

Simple harmonic motion is a repeated back-and-forth motion caused by a restoring force. This experiment studies two common oscillating systems: a simple pendulum and a vertical spring. The pendulum part investigates how amplitude, length, and bob mass affect the period. The spring part investigates how the supported mass affects the period of vertical oscillation.

Target Learning Outcome

TLO 7: Describe the relationships between period and mass in a vertical spring system; and between period and length, amplitude, and mass in a simple pendulum system.

I. Discussion of Theory

Equilibrium Position

Equilibrium position is the resting position of an object where the net force acting on it is zero.

Simple Harmonic Motion

Simple harmonic motion, or SHM, is periodic motion in which the restoring force is directly proportional to displacement and directed toward the equilibrium position.

Restoring force condition

An ideal SHM system has a restoring force proportional to displacement but opposite in direction.

F = -kx

Variables

Symbol	Description	Unit
$F$	restoring force	N
$k$	spring constant or force constant	N/m
$x$	displacement from equilibrium	m

Amplitude

Amplitude is the maximum displacement of an oscillating object from its equilibrium position. In the pendulum part of this experiment, amplitude may be measured as the horizontal displacement from the rest position to the release position.

Small-Angle Approximation

The theoretical equation for the period of a simple pendulum assumes a small angular displacement, not merely a small horizontal amplitude. For small angles, the angular displacement $\theta$ in radians can be approximated if the amplitude $A$ and length $L$ are known:

\theta \approx \frac{A}{L}

Period

The period, $T$ , is the time required to complete one full vibration or cycle.

Period from 10 vibrations

Timing 10 vibrations reduces the effect of stopwatch reaction time.

T = \frac{t_{10}}{10}

Variables

Symbol	Description	Unit
$T$	period of one vibration	s
$t_{10}$	total time for 10 vibrations	s

Frequency

Frequency is the number of vibrations completed per second. It is the reciprocal of the period.

Frequency

f = \frac{1}{T}

Variables

Symbol	Description	Unit
$f$	frequency	$Hz or s^-1$

Simple pendulum period

For small angular displacements, the period of a simple pendulum depends mainly on length and gravitational acceleration.

T = 2\pi\sqrt{\frac{L}{g}}

Variables

Symbol	Description	Unit
$L$	length of the pendulum from support point to center of bob	m
$g$	acceleration due to gravity	$m/s^2$

Pendulum relationships

For small amplitudes, the period of a simple pendulum is nearly independent of amplitude and mass. The period increases as the square root of the length:

T \propto \sqrt{L}

A longer pendulum swings more slowly and has a longer period.

Vertical spring period

For a mass suspended from an ideal spring, the period depends on the oscillating mass and the spring constant.

T = 2\pi\sqrt{\frac{M}{k}}

Variables

Symbol	Description	Unit
$M$	oscillating mass supported by the spring	kg
$k$	spring constant	N/m

Spring graph relationship

Squaring the spring-period equation gives a linear relationship between period squared and mass.

T^2 = \frac{4\pi^2}{k}M

If the graph uses $T^2$ as the horizontal axis and mass $M$ as the vertical axis, then:

M = \frac{k}{4\pi^2}T^2

Variables

Symbol	Description	Unit
$T^2$	period squared	$s^2$
$M$	effective oscillating mass	kg

For the purposes of this lab, $M$ represents the effective oscillating mass.

If $M$ is plotted vs $T^2$ , the slope of the resulting straight line is:

\text{slope} = \frac{k}{4\pi^2}

Therefore, the spring constant $k$ can be determined from the slope:

k = 4\pi^2(\text{slope})

Alternatively, if $T^2$ is plotted vs $M$ , the slope is $\frac{4\pi^2}{k}$ , and therefore:

k = \frac{4\pi^2}{\text{slope}}

Interpreting the vertical intercept

If the graph plots mass on the pan, $M$ , as the ordinate and $T^2$ as the abscissa, the vertical intercept may be negative. The magnitude of this intercept represents the effective mass already present in the system, such as the pan and part of the spring, depending on how the table defines $M$ .

II. Equipment / Materials Needed

Equipment or material	Purpose
1-meter string	Used to make a simple pendulum.
Two balls with different masses	Used as pendulum bobs for mass comparison.
Set of weights	Added to the spring pan to vary oscillating mass.
Pan for supporting weights	Holds masses attached to the spring.
Cylindrical or helical spring	Used for vertical spring oscillation.
Iron stand	Supports the pendulum and spring.
Meterstick	Measures string length and amplitude.
Stopwatch	Measures time for 10 vibrations.

Safety and setup reminders

Make sure the stand is stable before releasing the pendulum or spring. Use small amplitudes for the pendulum and small vertical displacements for the spring. Do not overstretch the spring.

III. Diagram of Setup

Simple pendulum setup

          support
            |
            |
            |  L
            |
           (o) bob

Measure L from the support point to the center of the bob.
Release from a small amplitude without pushing.

Graph of M vs T² concept

   M (mass)
   |
   |       /
   |      /
   |     /  slope = k / (4\pi²)
   |    /
   |   /
   |  /
   | /
---|----------------- T² (period squared)
   |
   |  y-intercept (effective mass of pan/spring)

A straight line indicates that $M$ is directly proportional to $T^2$ . If the effective mass of the pan/spring is accounted for, the line will pass through the origin.

Vertical spring setup

          iron stand
             |
          /\/\/\/\   spring
             |
            [ ]      pan + masses
             |
        vertical motion

Pull the pan slightly downward and release.
Count 10 complete up-and-down cycles.

IV. Procedures

A. Pendulum: effect of amplitude

STEP-BY-STEP

Attach one ball to a string and tie the other end of the string to the iron stand to form a simple pendulum.
Make sure the ball can swing freely without touching the stand or table.
Keep the length of the string constant.
Displace the ball by a small measured amplitude and release it without pushing.
Measure the total time for 10 vibrations.
Repeat for at least three different amplitudes.
Compute the period using $T=t_{10}/10$ .
Compute the frequency using $f=1/T$ .
Record the results in Table 7.1.
Describe the relationship between period and amplitude.

B. Pendulum: effect of length

STEP-BY-STEP

Keep the amplitude approximately constant.
Set the pendulum to a chosen length and measure the length from the support point to the center of the ball.
Release the ball from the same small amplitude without pushing.
Measure the total time for 10 vibrations.
Repeat for at least three different string lengths.
Compute the period and frequency for each length.
Record the results in Table 7.2.
Describe the relationship between period and length.

C. Pendulum: effect of mass

STEP-BY-STEP

Keep both the length and amplitude constant.
Measure the time for 10 vibrations using the first ball.
Replace the first ball with a second ball of different mass.
Use the same length and amplitude as before.
Measure the time for 10 vibrations using the second ball.
Compute the period and frequency for each ball.
Record the results in Table 7.3.
Describe the relationship between period and mass.

D. Vertical spring: effect of mass

STEP-BY-STEP

Attach the pan to the lower end of the helical spring.
Mount the upper end of the spring securely on the iron stand.
Record the mass of the pan and the mass of the spring.
Add a known mass to the pan.
Pull the spring straight downward through a small displacement.
Release the spring so the pan moves vertically up and down.
Count 10 complete cycles. One cycle is completed when the pan returns to its starting position with the same direction of motion.
Record the time for 10 cycles.
Repeat with increasing masses on the pan.
Compute $T$ , $T^2$ , and $f$ for each mass.
Record the results in Table 7.4.
Plot mass on the pan, $M$ , as the ordinate and $T^2$ as the abscissa.
Draw a straight line through the mean of the points and extend it until it intersects the vertical axis.
Interpret the relationship between mass and $T^2$ and explain the meaning of the vertical intercept.

Counting vibrations

Start timing when the object passes a clear reference position. Count 10 complete cycles, not 10 half-cycles. For consistency, stop timing when the object returns to the same position moving in the same direction.

V. Student Information

Field	Entry
Name
Schedule
Group No.
Date Performed

VI. Data and Results

Table 7.1. Simple Pendulum with Constant Length

Constant length: $L =$ ______ cm

Trial	Amplitude, $A$ (cm)	Time for 10 vibrations, $t_{10}$ (s)	Period, $T$ (s)	Frequency, $f$ (Hz)
1
2
3

Relationship between period and amplitude:

Observation prompt	Response
Does the period change significantly when amplitude changes within the small-angle range?

Table 7.2. Simple Pendulum with Constant Amplitude

Constant amplitude: $A =$ ______ cm

Trial	Length, $L$ (cm)	Time for 10 vibrations, $t_{10}$ (s)	Period, $T$ (s)	Frequency, $f$ (Hz)
1
2
3

Relationship between period and length:

Observation prompt	Response
What happens to the period as the string length increases?

Table 7.3. Simple Pendulum with Different Bob Masses

Constant amplitude: $A =$ ______ cm
Constant length: $L =$ ______ cm

Ball	Approximate mass	Time for 10 vibrations, $t_{10}$ (s)	Period, $T$ (s)	Frequency, $f$ (Hz)
Ball 1
Ball 2

Relationship between period and mass:

Observation prompt	Response
Does changing the bob mass significantly change the period when length and amplitude are constant?

Table 7.4. Helical Spring Supporting Different Weights

Mass of pan: ______ g
Mass of spring: ______ g

Trial	Mass on pan, $M$ (g)	Time for 10 cycles, $t_{10}$ (s)	Period, $T$ (s)	Period squared, $T^2$ (s²)	Frequency, $f$ (Hz)
1
2
3
4
5

Spring Graphing Task

Plot $T^2$ as the abscissa and the mass on the pan, $M$ , as the ordinate. Draw a straight line through the mean of the points and extend the line until it intersects the vertical axis.

Graph question	Response
Describe the relationship between mass and $T^2$ .
Interpret the value of the vertical intercept.
What does the slope suggest about the spring constant?

VII. Computations

Required computations

STEP-BY-STEP

Compute $T=t_{10}/10$ for every pendulum and spring trial.
Compute $f=1/T$ for every trial.
For the spring data, compute $T^2$ for every mass.
For the pendulum length test, compare measured $T$ with the trend $T\propto\sqrt{L}$ .
For the spring test, plot $M$ versus $T^2$ .
Determine whether the spring graph is approximately linear.
Interpret the vertical intercept of the spring graph.
State which variables significantly affect the period of a simple pendulum and a spring oscillator.

Sample period and frequency computation

If the time for 10 vibrations is $18.6\,\text{s}$ , then:

T = \frac{t_{10}}{10} = \frac{18.6}{10} = 1.86\,\text{s}

f = \frac{1}{T} = \frac{1}{1.86} = 0.538\,\text{Hz}

Sample spring graph computation

If $T = 1.20\,\text{s}$ , then:

T^2 = (1.20)^2 = 1.44\,\text{s}^2

This value is plotted on the horizontal axis. The corresponding mass on the pan is plotted on the vertical axis.

VIII. Expected Trends

Pendulum Period vs. Amplitude: For small amplitudes, changing the amplitude has very little to no effect on the period of the pendulum.
Pendulum Period vs. Length: A longer pendulum will have a longer period. The period is proportional to the square root of the length.
Pendulum Period vs. Mass: The mass of the pendulum bob has little to no effect on the period.
Spring Period vs. Mass: A larger mass on the spring results in a longer period.
Graph of M vs. T²: The plot of mass versus period squared should result in an approximately linear relationship.

IX. Error Analysis

Common sources of error

Stopwatch reaction time when starting and stopping measurements.
Counting half-cycles instead of full cycles.
Releasing the pendulum or spring with a push.
Using too large a pendulum amplitude, which violates the small-angle approximation.
Measuring pendulum length to the bottom of the bob instead of to its center.
Air resistance and friction at the support point.
Spring motion not purely vertical.
Spring stretched beyond its elastic range.
Masses on the pan shifting during oscillation.

Ways to improve accuracy

Time 10 or more complete vibrations instead of only one vibration.
Use small amplitudes for the pendulum.
Release without pushing.
Repeat trials and average the measured times.
Measure pendulum length from the pivot to the center of the bob.
Keep spring oscillations vertical and small.
Use the same reference point for counting cycles.

X. Observations and Conclusions

Conclusion guide

A strong conclusion should describe which variables affected the period and which did not. For the pendulum, state that length has the strongest effect on period for small amplitudes, while bob mass has little effect. For the spring system, state that increasing the supported mass increases the period and that mass is approximately linearly related to $T^2$ .

XI. Questions and Problems

Define simple harmonic motion, amplitude, period, and frequency.
Why is it better to measure the time for 10 vibrations instead of only one vibration?
A pendulum completes 10 vibrations in $16.0\,\text{s}$ . Find its period and frequency.
According to the small-angle pendulum equation, what happens to the period when the length is quadrupled?
Does the mass of the pendulum bob affect the period? Explain using the formula and your data.
Why should the pendulum be released from a small amplitude?
A vertical spring system has a larger mass added to the pan. What happens to the period? Explain using the spring-period formula.
What does the vertical intercept of the $M$ versus $T^2$ graph suggest about the spring system?

XII. Lab Report Format

Your lab report should contain the following sections:

Title Page: Experiment title, your name, group members, and date.
Objectives: Brief statement of the experiment's goals.
Theory: Key concepts and equations used.
Apparatus: List of equipment.
Procedure: Summary of steps taken.
Data and Results: Completed tables and the required graph.
Computations: Sample calculations for period, frequency, and spring constant.
Error Analysis: Discussion of potential errors and their impact.
Conclusions: Summary of findings based on the expected trends.
Answers to Questions: Responses to the post-lab questions.

XIII. Selected Answer Key

Question 3: $T = 16.0/10 = 1.60\text{ s}$ , $f = 1/1.60 = 0.625\text{ Hz}$
Question 4: If the length is quadrupled, the period is doubled (since $T \propto \sqrt{L}$ ).

XIV. References

Bueche, F. J., & Hecht, E. (1997). Schaum's Outline of Theory and Problems of College Physics (9th ed.). New York: McGraw-Hill.

Instructor note

The original HTML worksheet was converted into MDX and expanded with SHM theory, formulas, setup diagrams, procedures, data tables, graphing instructions, computation examples, error analysis, conclusion prompts, and post-lab questions. HTML input fields and theme controls were converted into printable MDX tables and prompts for compatibility with the CE content renderer.

Previous TopicLab 06: Archimedes' Principle and Specific Gravity

Quiz Me

Next TopicLab 08: Moment of Inertia and Torque

Prev Next

Quiz Me