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Introduction to the Model

Description and Assumptions

This model applies to position of a single point particle, or to the angular position of a rigid body, which is constrained to one dimension and experiences a restoring force that is linearly proportional to its displacement from an equilibrium position. This form for the force or torque implies that the equation of motion for the point particle or rigid body will have the form:

\begin{large}\[ a = \frac{d^{2}x}{dt^{2}} = - \omega_{\rm osc}^{2}x \]\end{large}

or

\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega_{\rm osc}^{2}\theta.\] \end{large}

As a consequence of this characteristic equation, the position, velocity, and acceleration (or the angular equivalents) will each be sinusoidal functions of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).

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).

Relevant Definitions

\begin{large}\[ A \equiv x_{\rm max} = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega_{\rm osc}}\right)^{2}} \]\end{large}

\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega_{\rm osc} A}\right)\]\end{large}

S.I.M. Structure of the Model

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.

Laws of Change

Mathematical Representation

\begin{large}\[ x(t) = x_{i}\cos(\omega_{\rm osc} (t-t_{i})) + \frac{v_{i}}{\omega_{\rm osc}}\sin(\omega_{\rm osc} (t-t_{i}))\]\end{large}

\begin{large}\[ v(t) = -\omega_{\rm osc} x_{i}\sin(\omega_{\rm osc} (t-t_{i})) + v_{i}\cos(\omega_{\rm osc} (t-t_{i}))\]\end{large}

\begin{large}\[ a(t) = -\omega_{\rm osc}^{2} x_{i}\cos(\omega_{\rm osc} (t-t_{i})) - \omega_{\rm osc} v_{i} \sin(\omega_{\rm osc} (t-t_{i})) = -\omega_{\rm osc}^{2} x(t) \]\end{large}

\begin{large}\[ x(t) = A\cos(\omega_{\rm osc} t + \phi)\]\end{large}

\begin{large}\[ v(t) =-A\omega_{\rm osc}\sin(\omega_{\rm osc} t + \phi)\]\end{large}

\begin{large}\[ a(t) =-\omega_{\rm osc}^{2}A\cos(\omega_{\rm osc} t+\phi) =-\omega_{\rm osc}^{2} x(t)\]\end{large}

Diagrammatic Representations

• position versus time graph
• velocity versus time graph
• acceleration versus time graph

Relevant Examples

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id Pend

Examples involving Pendulums