Experiment 7: Simple Harmonic Motion - Theory & Concepts

Experiment 7: Simple Harmonic Motion

Learning Objectives

Define periodic motion, equilibrium, displacement, amplitude, phase, period, frequency, angular frequency, restoring force, damping, resonance, and simple harmonic motion.
Explain the mathematical condition that distinguishes simple harmonic motion from general periodic motion.
Relate displacement, velocity, acceleration, force, and energy throughout one oscillation cycle.
Analyze ideal and real mass-spring systems, including effective mass and damping.
Explain the simple-pendulum model, the small-angle approximation, and the variables that affect pendulum period.
Interpret displacement-time, velocity-time, acceleration-time, energy, and linearized experimental graphs.
Use interactive simulations to test theoretical predictions before performing the laboratory activity.

Simple harmonic motion is one of the most important mathematical models in physics. It describes systems that oscillate about a stable equilibrium when the restoring force is proportional to displacement. Springs, small-angle pendulums, vibrating structural members, vehicle suspensions, seismometers, and many sensor systems can be understood by extending this model.

Target Learning Outcome

Explain and predict the relationships among period, frequency, mass, stiffness, pendulum length, amplitude, damping, and gravitational acceleration in oscillating systems.

1. Foundations of Oscillatory Motion

Periodic Motion

Periodic motion is motion that repeats after equal time intervals. Every simple harmonic motion is periodic, but not every periodic motion is simple harmonic.

Equilibrium Position

The equilibrium position is the position where the net force is zero. A stable equilibrium produces forces that tend to return a displaced object toward that position.

Displacement

Displacement $x$ is the signed distance of the oscillator from equilibrium. Positive and negative signs identify opposite sides of the equilibrium position.

Amplitude

Amplitude $A$ is the maximum magnitude of displacement from equilibrium. It describes the size of the oscillation.

Cycle

A cycle is one complete repetition of motion in which the oscillator returns to the same position while moving in the same direction.

Period

The period $T$ is the time required for one complete cycle. Its SI unit is the second.

Frequency

Frequency $f$ is the number of complete cycles per second. Its SI unit is the hertz, where $1\,\text{Hz}=1\,\text{s}^{-1}$ .

Angular Frequency

Angular frequency $\omega$ is the rate of change of phase, measured in radians per second. One complete cycle corresponds to $2\pi$ radians.

Period, frequency, and angular frequency

These quantities describe the same oscillation rate in different forms.

f=\frac{1}{T}

\omega=2\pi f=\frac{2\pi}{T}

Variables

Symbol	Description	Unit
$T$	period	s
$f$	frequency	Hz
$\omega$	angular frequency	rad/s

Phase

Phase specifies the state of an oscillator within its cycle. Two oscillators with equal frequency can still be at different positions and moving in different directions because their phases differ.

How position changes during one cycle

At the extreme positions $x=\pm A$ , the oscillator momentarily stops before reversing direction. At equilibrium $x=0$ , the speed is maximum. The restoring force and acceleration always point toward equilibrium, not necessarily in the direction of motion.

2. The Defining Condition for Simple Harmonic Motion

Restoring Force

A restoring force is a force directed toward a stable equilibrium position. In ideal simple harmonic motion, its magnitude is proportional to displacement.

Simple Harmonic Motion

Simple harmonic motion is oscillatory motion in which acceleration is proportional to displacement and opposite in direction.

Restoring-force condition for simple harmonic motion

The negative sign indicates that the force opposes displacement.

F=-kx

Variables

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

Equation of motion for a mass-spring oscillator

Newton's second law converts the restoring-force relation into a differential equation.

m\frac{d^2x}{dt^2}+kx=0

\frac{d^2x}{dt^2}=-\frac{k}{m}x=-\omega^2x

Variables

Symbol	Description	Unit
$m$	oscillating mass	kg
$x$	displacement from equilibrium	m
$k$	spring stiffness	N/m
$\omega$	natural angular frequency	rad/s

Diagnostic test for simple harmonic motion

A motion is simple harmonic only when the acceleration-displacement graph is a straight line through the origin with negative slope. A repeating motion that does not satisfy $a=-\omega^2x$ is periodic but not ideal simple harmonic motion.

3. Kinematics and Phase Relationships

Sinusoidal displacement

The general displacement function for ideal simple harmonic motion.

x(t)=A\cos(\omega t+\phi)

Variables

Symbol	Description	Unit
$x(t)$	instantaneous displacement	m
$A$	amplitude	m
$\omega$	angular frequency	rad/s
$t$	time	s
$\phi$	phase constant	rad

Velocity and acceleration in simple harmonic motion

Velocity is one-quarter cycle out of phase with displacement, while acceleration is opposite to displacement.

v(t)=-A\omega\sin(\omega t+\phi)

a(t)=-A\omega^2\cos(\omega t+\phi)=-\omega^2x(t)

Variables

Symbol	Description	Unit
$v(t)$	instantaneous velocity	m/s
$a(t)$	instantaneous acceleration	$m/s^2$

Maximum speed and acceleration

The maximum values depend on amplitude and angular frequency.

v_{\text{max}}=A\omega

a_{\text{max}}=A\omega^2

Variables

Symbol	Description	Unit
$v_{\text{max}}$	maximum speed	m/s
$a_{\text{max}}$	maximum acceleration magnitude	$m/s^2$

Position, velocity, acceleration, and force at key locations

Location	Displacement	Speed	Acceleration magnitude	Energy condition
Positive extreme	$+A$	$0$	maximum	potential maximum
Equilibrium	$0$	maximum	$0$	kinetic maximum
Negative extreme	$-A$	$0$	maximum	potential maximum

4. Energy in an Ideal Oscillator

Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy. In an ideal undamped oscillator, total mechanical energy remains constant.

Energy of a mass-spring oscillator

Energy continually changes form while the ideal total remains constant.

E=\frac{1}{2}kA^2

U=\frac{1}{2}kx^2

K=E-U=\frac{1}{2}k\left(A^2-x^2\right)

Variables

Symbol	Description	Unit
$E$	total mechanical energy	J
$U$	elastic potential energy	J
$K$	kinetic energy	J

Energy interpretation

At the extremes, the oscillator has maximum potential energy and zero kinetic energy. At equilibrium, the potential energy is minimum and kinetic energy is maximum. Energy conservation explains why the oscillator speeds up toward equilibrium and slows down toward an extreme.

5. Mass-Spring Systems

Spring Constant

The spring constant $k$ measures stiffness. A larger $k$ means that a greater force is required to produce the same deformation.

Natural frequency and period of a mass-spring oscillator

The oscillation rate depends on inertia and stiffness.

\omega_n=\sqrt{\frac{k}{m}}

f_n=\frac{1}{2\pi}\sqrt{\frac{k}{m}}

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

Variables

Symbol	Description	Unit
$\omega_n$	natural angular frequency	rad/s
$f_n$	natural frequency	Hz
$T$	period	s
$m$	oscillating mass	kg
$k$	spring constant	N/m

Effect of changing mass, stiffness, and amplitude

Increasing mass increases period according to $T\propto\sqrt{m}$ .
Increasing stiffness decreases period according to $T\propto1/\sqrt{k}$ .
In ideal linear motion, amplitude changes energy but does not change period.
Gravity shifts the equilibrium position of a vertical spring but does not appear in the period measured about that equilibrium.

Effective Oscillating Mass

Effective oscillating mass is the total inertia participating in the motion. It can include the added load, hanger, attachments, and part of the spring mass.

Effective mass of a vertical spring setup

A common introductory approximation includes one-third of the mass of a uniform spring.

m_{\text{eff}}=m_{\text{load}}+m_{\text{hanger}}+\frac{1}{3}m_{\text{spring}}

Variables

Symbol	Description	Unit
$m_{\text{eff}}$	effective oscillating mass	kg
$m_{\text{load}}$	added load mass	kg
$m_{\text{hanger}}$	hanger or pan mass	kg
$m_{\text{spring}}$	spring mass	kg

Interactive spring simulation

Change mass, stiffness, amplitude, and damping. Predict the effect on period before moving each control, then compare the prediction with the displayed response.

Interactive engineering simulation

Simple Harmonic Motion & Damping

Study the oscillatory behavior of a mass-spring system. Introduce damping to see how the system transitions from standard oscillation to critical and overdamping.

Mass (m)1.0 kg

0.5–5.0

Drag for exploration or enter an exact value. Press Enter to apply; Escape restores the current value.

Spring Constant (k)10 N/m

N/m

2–50

Drag for exploration or enter an exact value. Press Enter to apply; Escape restores the current value.

Initial Amplitude (x0)0.5 m

0.1–1.5

Drag for exploration or enter an exact value. Press Enter to apply; Escape restores the current value.

Damping Coefficient (c)0.0 kg/s

kg/s

0.0–5.0

Drag for exploration or enter an exact value. Press Enter to apply; Escape restores the current value.

Damping Ratio (ζ) & State

0.00Undamped

Governing Formulas

Motion Equation

m \ddot{x} + c \dot{x} + k x = 0

Natural Angular Freq

\omega_0 = \sqrt{k/m}, \quad \zeta = \frac{c}{2\sqrt{km}}

Natural Freq0.50 Hz

Period (T)1.99 s

Position (x)0.50 m

Velocity (v)0.00 m/s

Model scope and verification

Use the displayed units and idealizations, then verify the governing balance or compatibility equation before interpreting the result.

6. Damping, Driven Motion, and Resonance

Damping

Damping is the loss of mechanical energy due to mechanisms such as friction, air resistance, fluid drag, or internal material hysteresis.

Natural Frequency

Natural frequency is the frequency at which a system tends to oscillate after a small disturbance when no continuing external periodic force acts.

Driven Oscillation

A driven oscillation occurs when a periodic external force continuously supplies energy to an oscillator.

Resonance

Resonance is the large response that can occur when the driving frequency is close to a system's natural frequency.

Damped mass-spring equation

A viscous damping force proportional to velocity modifies the ideal oscillator equation.

m\frac{d^2x}{dt^2}+c\frac{dx}{dt}+kx=0

Variables

Symbol	Description	Unit
$c$	viscous damping coefficient	N·s/m

Three damping regimes

Underdamped: oscillation continues while amplitude decays.
Critically damped: the system returns to equilibrium as quickly as possible without oscillating.
Overdamped: the system returns without oscillating but more slowly than the critically damped case.

Engineering significance of resonance

Resonance can amplify motion in bridges, buildings, machinery, and vehicle systems. Engineers control it by changing mass or stiffness, adding damping, isolating the source, or avoiding operating frequencies near resonance.

7. Simple Pendulum Theory

Simple Pendulum

A simple pendulum is an ideal point mass suspended from a fixed support by a massless, inextensible string. Its length is measured from the pivot to the center of mass of the bob.

Exact angular equation of a pendulum

The sine term makes the exact pendulum nonlinear.

\frac{d^2\theta}{dt^2}+\frac{g}{L}\sin\theta=0

Variables

Symbol	Description	Unit
$\theta$	angular displacement	rad
$g$	gravitational acceleration	$m/s^2$
$L$	pendulum length	m

Small-Angle Approximation

The small-angle approximation replaces $\sin\theta$ with $\theta$ when $\theta$ is expressed in radians and is sufficiently small.

Small-angle pendulum model

Linearization converts the nonlinear pendulum into a simple harmonic oscillator.

\sin\theta\approx\theta

\frac{d^2\theta}{dt^2}+\frac{g}{L}\theta=0

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

Variables

Symbol	Description	Unit
$T$	small-angle period	s

Variables affecting pendulum period

Increasing length increases period according to $T\propto\sqrt{L}$ .
Increasing gravitational acceleration decreases period according to $T\propto1/\sqrt{g}$ .
Bob mass does not appear in the ideal period formula.
Small changes in amplitude have little effect.
Large release angles produce periods longer than the small-angle prediction.

Limits of the ideal pendulum model

The simple formula assumes a small angle, fixed pivot, light nonstretching string, compact bob, negligible air resistance, and planar motion. Measuring length to the top or bottom of the bob instead of its center introduces systematic error.

Interactive pendulum simulation

Vary length, bob mass, release angle, and damping. Test the prediction that mass does not control the ideal period, while length and large-angle behavior do.

Simple Pendulum — Exact Numerical Integration

Pendulum Setup

Length (

L

)1.5 m

Mass (

m

)2.0 kg

Initial Angle (

\theta_0

)30°

Damping Ratio (

\zeta

)0.10

Dynamics Equations

Equation of Motion:

\ddot{\theta} + \frac{g}{L}\sin\theta = 0

Small-Angle Period:

T = 2\pi\sqrt{\frac{L}{g}} = 2.457\,\text{s}

θ (current):30.0°

\omega

:0.000 rad/s

T

(current):0.00 s

Normal (

N

):16.99 N

Energy Balance

PE3.94 J

KE0.00 J

Total E₀3.94 J

Phase Portrait (θ, ω)

8. Graphs and Experimental Linearization

Period from repeated cycles

Timing many cycles reduces the relative effect of reaction time.

T=\frac{t_N}{N}

Variables

Symbol	Description	Unit
$t_N$	measured time for N cycles	s
$N$	number of cycles	cycle

Pendulum linearization

Squaring the period relation produces a straight-line form.

T^2=\frac{4\pi^2}{g}L

Variables

Symbol	Description	Unit
$T^2$	period squared	$s^2$
$L$	pendulum length	m

Spring linearization

A period-squared graph can determine spring stiffness and effective-mass effects.

T^2=\frac{4\pi^2}{k}m_{\text{eff}}

Variables

Symbol	Description	Unit
$T^2$	period squared	$s^2$
$m_{\text{eff}}$	effective oscillating mass	kg

How to interpret the principal graphs

Graph	Ideal shape	Physical meaning
$x$ versus $t$	sinusoidal	displacement history
$v$ versus $t$	sinusoidal, shifted by one-quarter cycle	velocity history
$a$ versus $t$	opposite phase to displacement	restoring acceleration
$a$ versus $x$	straight line with negative slope	SHM diagnostic
$T^2$ versus $L$	straight line	pendulum determination of $g$
$T^2$ versus $m_{\text{eff}}$	straight line	spring determination of $k$

9. Laboratory Application

Theory-guided investigation

STEP-BY-STEP

Use the simulations to form predictions about mass, length, stiffness, amplitude, and damping.
Measure the period by timing at least $10$ complete cycles.
Repeat measurements and use mean values.
For the pendulum, vary one factor at a time while controlling the others.
For the spring, vary effective mass while keeping oscillations small and vertical.
Construct the assigned linearized graph and determine its slope.
Compare the measured relationships with the theoretical proportionalities.
Explain discrepancies using model limitations and measurement uncertainty.

Common experimental errors

Incorrect cycle counting
Stopwatch reaction time
Release with an unintended push
Large pendulum angle
Incorrect pendulum-length reference
Elliptical pendulum motion
Sideways spring motion
Neglect of hanger or spring mass
Spring deformation beyond the elastic range
Use of individual point-to-point slopes instead of a best-fit line

Engineering connections

Oscillation theory supports earthquake engineering, vibration isolation, tuned mass dampers, bridge and floor serviceability, machine foundations, suspension design, instrumentation, and structural health monitoring. The laboratory systems are simple, but the same ideas of inertia, stiffness, damping, natural frequency, and resonance govern much larger engineering systems.

Key Takeaways

Simple harmonic motion requires acceleration proportional and opposite to displacement.
Period, frequency, and angular frequency satisfy $f=1/T$ and $\omega=2\pi/T$ .
Displacement, velocity, and acceleration are sinusoidal but differ in phase.
Energy alternates between kinetic and potential forms in an ideal oscillator.
A mass-spring system has $T=2\pi\sqrt{m/k}$ .
A small-angle pendulum has $T=2\pi\sqrt{L/g}$ .
Pendulum mass does not determine the ideal period.
Damping removes energy, while resonance can greatly amplify response.
Linearized graphs reveal physical constants and help test theoretical models.
Simulations are most useful when students predict first, vary one factor at a time, and explain the observed trend.

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