Resource Lesson
Simple Harmonic Motion
Printer Friendly Version
Suppose that an oscillating spring has one end firmly attached to a base of support and a mass attached to its free end. As the mass vibrates back and forth, we can track the behavior of three instantaneous quantities: the mass' displacement, velocity, and acceleration. Note in the diagrams shown below that when the mass' displacement is at a maximal positive position, its velocity is zero, and its acceleration, which is acting to restore the mass to its undisturbed equilibrium position, has a maximum negative value.
Notice that at the endpoints, when v = 0, the mass has no kinetic energy, KE = ½mv
^{2}
. Therefore, all of its energy is in the form of elastic potential energy, PE
_{e}
= ½kx
^{2}
. When PE
_{e}
is maximum, the restoring force within the spring is also maximized resulting in the mass' acceleration also being maximized as the spring acts to return the mass to its equilibrium position.
There are two formulas at our disposal to quantify the restoring force within the spring as it oscillates: Newton's 2nd Law, net F = ma, and Hooke's Law, F = - ks:
This results tells us that the mass' instantaneous acceleration is directly proportional to, but in the opposite direction as, its instantaneous displacement. To help us understand the substitution which we will need to use next, we are going to return to some relationships which we learned for uniform circular motion. In the following video, note how the motion of the ball's shadow emulates the motion of a mass on the end of a vibrating spring.
SHM
position
Additional videos and a physlet animations that will clarify further the relationships between position, velocity and acceleration are provided in the chart below.
SHM
Physlet Animation
velocity
Circular/SHM
acceleration
sinusoidal wave
In general, the
sinusoidal equations
for each property graphed at the top of this page are summarized in the following equations.
where
represents the frequency measured in hertz
and ω, or the angular velocity, equals
measured in rad/sec
Returning to our previous result of
please note:
that the magnitude of a
_{max}
is equivalent to the magnitude of the mass' centripetal acceleration, a
_{c}
= v
^{2}
/r
the fact that s
_{max}
equals the radius of the circle,
r
, or the amplitude,
A
, of the sine graph
Remembering the relationship from circular motion that
we can substitute
A
for
r
and complete our derivation.
Summary of SHM
The following list summarizes the properties of simple harmonic oscillators.
The oscillator's motion is
periodic;
that is, it is repetitive at a constant frequency.
The
restoring force
within the oscillating system is proportional to the negative of the oscillator's displacement and acts to restore it to equilibrium.
The
velocity
of the oscillator is maximum as it passes through equilibrium, and zero as it passes through the extreme positions in its oscillation.
The
acceleration
experienced by the oscillator is proportional to the negative of its displacement from the midpoint of its motion.
Related Documents
Lab:
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Calculation of "g" Using Two Types of Pendulums
Labs -
Conical Pendulums
Labs -
Conical Pendulums
Labs -
Conservation of Energy and Vertical Circles
Labs -
Directions: Constructive and Destructive Interference
Labs -
Doppler Effect: Source Moving
Labs -
Frequency of Vibrating Strings
Labs -
Illuminance by a Light Source
Labs -
Inertial Mass
Labs -
Interference Shading
Labs -
Introductory Simple Pendulums
Labs -
Kepler's 1st and 2nd Laws
Labs -
Loop-the-Loop
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Oscillating Springs
Labs -
Pipe Music
Labs -
Relationship Between Tension in a String and Wave Speed
Labs -
Relationship Between Tension in a String and Wave Speed Along the String
Labs -
Ripple Tank Checklists
Labs -
Ripple Tank Checklists
Labs -
Ripple Tank Sample Solutions
Labs -
Ripple Tank Student Involvement Sheet
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Sand Springs
Labs -
Simple Pendulums: Class Data
Labs -
Simple Pendulums: LabPro Data
Labs -
Speed of a Wave Along a Spring
Labs -
Speed of Sound in Air
Labs -
Speed of Sound in Copper
Labs -
Video LAB: A Gravitron
Labs -
Video LAB: Circular Motion
Labs -
Video LAB: Looping Rollercoaster
Labs -
Video: Law of Reflection
Labs -
Video: Law of Reflection Sample Diagram
Labs -
Water Springs
Resource Lesson:
RL -
A Derivation of the Formulas for Centripetal Acceleration
RL -
Barrier Waves, Bow Waves, and Shock Waves
RL -
Beats: An Example of Interference
RL -
Centripetal Acceleration and Angular Motion
RL -
Conservation of Energy and Springs
RL -
Derivation of Bohr's Model for the Hydrogen Spectrum
RL -
Derivation: Period of a Simple Pendulum
RL -
Energy Conservation in Simple Pendulums
RL -
Gravitational Energy Wells
RL -
Interference of Waves
RL -
Interference: In-phase Sound Sources
RL -
Introduction to Sound
RL -
Kepler's Laws
RL -
Law of Reflection
RL -
LC Circuit
RL -
Magnetic Forces on Particles (Part II)
RL -
Period of a Pendulum
RL -
Physical Optics - Thin Film Interference
RL -
Resonance in Pipes
RL -
Resonance in Strings
RL -
Ripple Tank Video Guides
RL -
Rotational Kinematics
RL -
SHM Equations
RL -
Sound Level Intensity
RL -
Speed of Waves Along a String
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
The Doppler Effect
RL -
The Law of Universal Gravitation
RL -
Thin Rods: Moment of Inertia
RL -
Uniform Circular Motion: Centripetal Forces
RL -
Universal Gravitation and Satellites
RL -
Vertical Circles and Non-Uniform Circular Motion
RL -
Vibrating Systems - Simple Pendulums
RL -
Vibration Graphs
RL -
Wave Fundamentals
RL -
Waveform vs Vibration Graphs
REV -
Orbitals
Review:
REV -
Chapter 26: Sound
REV -
Honors Review: Waves and Introductory Skills
REV -
Physics I Review: Waves and Introductory Skills
REV -
Review: Circular Motion and Universal Gravitation
REV -
Sound
REV -
Waves and Sound
REV -
Waves and Sound
Worksheet:
APP -
Big Al
APP -
Echo Chamber
APP -
Ring Around the Collar
APP -
The Dog-Eared Page
APP -
The Satellite
APP -
The Spring Phling
APP -
Timex
CP -
Centripetal Acceleration
CP -
Centripetal Force
CP -
Light Properties
CP -
Reflection
CP -
Satellites: Circular and Elliptical
CP -
Shock Waves
CP -
Sound
CP -
Waves and Vibrations
NT -
Apparent Depth
NT -
Atmospheric Refraction
NT -
Circular Orbits
NT -
Concert
NT -
Light vs Sound Waves
NT -
Pendulum
NT -
Rotating Disk
NT -
Shock Cone
NT -
Sound Waves
NT -
Spiral Tube
NT -
Standing Waves
WS -
Basic Practice with Springs
WS -
Beats
WS -
Beats, Doppler, Resonance Pipes, and Sound Intensity
WS -
Counting Vibrations and Calculating Frequency/Period
WS -
Doppler - A Challenge Problem
WS -
Doppler Effect
WS -
Fixed and Free-end Reflections
WS -
Fundamental Wave Terms
WS -
Illuminance 1
WS -
Illuminance 2
WS -
Inertial Mass Lab Review Questions
WS -
Interference: In-phase Sound Sources
WS -
Introduction to Springs
WS -
Kepler's Laws: Worksheet #1
WS -
Kepler's Laws: Worksheet #2
WS -
Lab Discussion: Inertial and Gravitational Mass
WS -
More Practice with Resonance in Pipes
WS -
More Practice with SHM Equations
WS -
More Practice with the Doppler Practice
WS -
Pendulum Lab Review
WS -
Pendulum Lab Review
WS -
Practice with Resonance in Pipes
WS -
Practice with the Doppler Effect
WS -
Practice: SHM Equations
WS -
Practice: Speed of a Wave Along a String
WS -
Practice: Uniform Circular Motion
WS -
Practice: Vertical Circular Motion
WS -
Pulse Superposition: Interference
WS -
Ripple Tank Review
WS -
SHM Properties
WS -
Sound Vocabulary
WS -
Speed of Sound
WS -
Speed of Sound (Honors)
WS -
Standing Wave Patterns #1
WS -
Standing Wave Patterns #2
WS -
Standing Wave Patterns #3
WS -
Standing Wave Patterns #4
WS -
Static Springs: The Basics
WS -
Universal Gravitation and Satellites
WS -
Vertical Circular Motion #1
WS -
Vibrating Systems - Period and Frequency
WS -
Wave Phenomena Reading Guide
WS -
Wave Pulses
WS -
Waveform and Vibration Graphs #1
WS -
Waveform and Vibration Graphs #2
TB -
25A: Introduction to Waves and Vibrations
TB -
25B: Vibrations and Waves
TB -
25C: Wave Speed
TB -
25D: Interference
TB -
25E: Doppler
TB -
25F: Doppler Effect (continued)
TB -
26B: Speed of Sound
TB -
26C: Resonance
TB -
26D: Beats
TB -
26E: Decibels
TB -
27A: Light Properties
TB -
Centripetal Acceleration
TB -
Centripetal Force
TB -
Decibels and Sound Intensity #1
TB -
Decibels and Sound Intensity #2
TB -
Interference Re-examined
TB -
Refraction Phenomena Reading Questions
TB -
Sound: Mixed Practice
TB -
Waves and Vibrations
PhysicsLAB
Copyright © 1997-2023
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton