Description and Assumptions
This model applies to position of a single point particle, or to the angle of a rigid body, constrained to move in one dimension that experiences a linear restoring force toward its equilibrium. Consequently, its position, velocity, and acceleration will each be 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 natural angular frequency ω0. The most common systems whose equations of motion take this form are a mass on a spring or a pendulum (in the small-angle approximation). Any problem requesting or giving position or velocity vs. time for one of these systems will likely require the use of the Simple Harmonic Motion model.
Another cue that Simple Harmonic Motion is occurring is if the position, the velocity, or the acceleration are sinusoidal in time (in which case they all will be sinusoidal).
Learning Objectives
Students will be assumed to understand this model who can:
- Define the terms equilibrium position and restoring force.
- Define the amplitude, period, natural angular frequency and phase of oscillatory motion.
- Give a formula for the natural angular frequency of the oscillation of a pendulum or mass on a spring.
- 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 natural 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 a stable equilibrium position.
Relevant Definitions
\begin
[ A = \sqrt{x_
^
+ \left(\frac{v_{i}}{\omega_{0}}\right)^{2}} ]\end
\begin
[ \phi = \cos^{-1}\left(\frac{x_{i}}
\right) = \sin^{-1}\left(\frac{v_{i}}{\omega_
A}\right)]\end
Laws of Change
\begin
[ x(t) = x_
\cos(\omega_
(t-t_
)) + \frac{v_{i}}{\omega_{0}}\sin(\omega_
(t-t_
))]\end
\begin
[ v(t) = -\omega_
x_
\sin(\omega_
(t-t_
)) + v_
\cos(\omega_
(t-t_
))]\end
\begin
[ a(t) = -\omega_
^
x_
\cos(\omega_
(t-t_
)) - \omega_
v_
\sin(\omega_
(t-t_
)) = -\omega_
^
x(t) ]\end
\begin
[ x(t) = A\cos(\omega_
t + \phi)]\end
\begin
[ v(t) =-A\omega_
\sin(\omega_
t + \phi)]\end
\begin
[ a(t) =-\omega_
^
A\cos(\omega_
t+\phi) =-\omega_
^
x(t)]\end
Diagrammatic Representations
- position versus time graph
- [velocity versus time graph]
- [acceleration versus time graph]
Relevant Examples
ExamplesinvolvingPendulums"> Examples involving Pendulums