Simple Harmonic Motion
Description and Assumptions
This model applies to a single point particle constrained to move in one dimension whose position is a sinusoidal function of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).
Problem Cues
Any object that experiences a linear restoring force or torque so that the equation of motion takes the form
\begin
[ a = \frac{d^
x}{dt^{2}} = - \omega^
x ]\end
or
\begin
[ \alpha = \frac{d^
\theta}{dt^{2}} = -\omega^
\theta] \end
will experience simple harmonic motion with angular frequency ω. The prototypical example is an object of mass m attached to a spring with force constant k, in which case, by Hooke's Law for elastic interactions:
\begin
[ a = -\frac
]\end
giving simple harmonic motion with angular frequency
$\sqrt{\dfrac
{m}}$
In the real world, most situations in which there is a stable position with a restoring force can be treated (for small oscillations at least) as if the restoring force is linear. Therefore, even systems with non-ideal springs and the oscillations of a pendulum can be treated as cases of Simple Harmonic Motion as long as the motion is small.
Another cue that Simple Harmonic Motion is occurring is if the poision, the velocity, or the acceleration are sinusoidal in time.
Learning Objectives
Students will be assumed to understand this model who can:
- Define the terms equilibrium position and restoring force.
- Define the amplitude, period, angular frequency and phase of oscillatory motion.
- Write mathematical expressions for the position, velocity and acceleration of Simple Harmonic Motion as functions of time for the special cases that the initial velocity is zero or the initial position is equilibrium.
- Graphically represent the position, velocity and acceleration of Simple Harmonic Motion.
- Use the laws of dynamics to determine the angular frequency of a system in the limit of very small displacements from equilibrium.
- Describe the consequences of conservation of mechanical energy for Simple Harmonic Motion (assuming no dissipation).
Models
Compatible Systems
A single point particle (or, for the angular version of SHM, a single rigid body).
Relevant Interactions
The system must be subject to a one-dimensional restoring force (or torque) that varies linearly with the displacement (or angular displacement) from an equilibrium position.
Relevant Definitions
Frequency
The frequency of oscillation of a simple oscillator is represent by a small Greek letter omega. For a simple mass m on a spring of constant k the frequency is given by
\begin
[ \omega = \sqrt{\frac
{m}} ]\end
For a simple pendulum of length L making small oscillations under the influence of gravity (with the acceleration due to gravity denoted by g) the frequency is
\begin
[ \omega = \sqrt{\frac
{L}} ]\end
Initial Conditions
Acceleration
\begin
[a_
= a(t = t_
) ]\end
Velocity
\begin
[v_
= v(t = t_
) ]\end
Position
\begin
[x_
= x(t = t_
) = -\frac{a_{i}}{\omega^{2}} ]\end
Amplitude of Motion
\begin
[ A = \sqrt{x_
^
+ \left(\frac{v_{i}}
\right)^{2}} ]\end
Phase
\begin
[ \phi = \cos^{-1}\left(\frac{x_{i}}
\right) = \sin^{-1}\left(\frac{v_{i}}
\right)]\end
Laws of Change
Position
\begin
[ x(t) = x_
\cos(\omega t) + \frac{v_{i}}
\sin(\omega t)]\end
or, equivalently
\begin
[ x(t) = A\cos(\omega t + \phi)]\end
Velocity
\begin
[ v(t) = -\omega x_
\sin(\omega t) + v_
\cos(\omega t)]\end
or, equivalently:
\begin
[ v(t) = -A\omega\sin(\omega t + \phi)]\end
Acceleration
\begin
[ a(t) = -\omega^
x_
\cos(\omega t) - \omega v_
\sin(\omega t) = -\omega^
x(t) ]\end
or, equivalently:
\begin
[ a(t) = -\omega^
A\cos(\omega t+\phi) = -\omega^
x(t)]\end
Diagrammatical Representations
- [acceleration versus time graph]
- [velocity versus time graph]
- position versus time graph
Relevant Examples
All Examples involving this Model