Simple Harmonic Motion

The single most important concept in the study of waves and sound is that of simple harmonic motion (SHM).

Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency.

Periodic MotionMotion that repeats in a regular pattern over and over again is called periodic motion. 

Simple harmonic motion is a specific type of periodic motion that has a simple sine or cosine wave shape.

Simple Harmonic Motion (SHM)Physicists like simple harmonic motion because every example of SHM is based on the same underlying physical principle and all examples of SHM have the same, very straightforward, mathematical description.

What is the physical principle?  SHM occurs around an equilibrium position when a mass is subject to a linear restoring force.  A linear restoring force is one that gets progressively larger with displacement from the equilibrium position.  The best example of this is a spring.  The more you stretch a spring the larger the force trying to get the spring back to its original shape.

Position VS. Time graph

What is the simple mathematical form of SHM motion? 

The displacement of the oscillating mass varies sinusoidally as a function of time. 

Periodic Motion Simple Harmonic Motion

Hearbeat Oscillating mass on a spring

HOOKE'S LAW

The restoring force of an ideal spring is given by,

where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in a direction opposite to the displacement of the spring.

Simple Harmonic Motion

When there is a restoring force, F = -kx, simple harmonic motion occurs.

Amplitude

Amplitude is the magnitude of the maximum displacement.

Period, T

For any object in simple harmonic motion, the time required to complete one cycle is the period T.

Frequency, f

The frequency f of the simple harmonic motion is the number of cycles of the motion per second.

Exercise on Simple Harmonic Motion

Q1. What is the amplitude?Q2. What is the period?Q3. What is the frequency?

Graphs of position, velocity, and acceleration

http://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.html