To Teach a Monkey (beta)

Simple Harmonic Motion

Simple Harmonic Motion is simply defined as motion whose acceleration is proportional to the displacement from equilibrium, and that the force always acts towards equilibrium

a∝−xa \propto -x


Terminology

Frequency

The number of cycles completed per second in SHM.

f=Frequency (Hz or s−1)f = \text{Frequency} \ (\text{Hz or s}^{-1})

Amplitude

The maximum displacement from the fixed equilibrium position.

A or x0=Amplitude (m)A \text{ or } x_0 = \text{Amplitude (m)}

Displacement

The distance of the object from the equilibrium position at a given time.

x=Displacement (m)x = \text{Displacement (m)}

Time Period

The time taken to complete one full cycle of SHM.

T=Time Period (s)T = \text{Time Period (s)}

Note T=1f=2πωT = \frac{1}{f} = \frac{2\pi}{\omega}

Angular Velocity

Angular velocity represents the velocity of the circular projection of the harmonic motion

Therefore we can represent the displacement as x=x0sin(ωt+shift)x = x_0 sin(\omega t + \text{shift})

Knowing how displacement is proportional to acceleration, we can write velocity as

v=ωx0sin(ωt+shift)v = \omega x_0 sin(\omega t + \text{shift})

and also

x=±ωx02−x2x = \pm \omega \sqrt{x_0^2 - x^2}

Phase Shift

Phase shift is the angle at which two oscillating objects differ by

Simple Harmonic Oscillators

Mass on a Spring

The period of oscillation for a mass on a spring is

T=2πmkT=2\pi \sqrt{\frac{m}{k}}

Pendulum

The period of oscillation for a pendulum is

T=2πLgT=2\pi \sqrt{\frac{L}{g}}