Today we’re covering:
 Simple Harmonic Motion
 Equations
 Graphs
 Examples of Simple Harmonic Motion
 Loaded Spring
 Pendulum
Let’s go!
What is Simple Harmonic Motion?
A type of oscillation where:
 Acceleration (a) is proportional to displacement (x)
 Acceleration (a) is always acting towards a fixed point / opposite in direction to the displacement
a ∝ x
The proportionality constant is the square of angular frequency:
a = ω^{2}x
Let’s see where this equation comes from by starting off with:
Representing Simple Harmonic Motion
As we have seen before, we can represent SHM with:
Sinusoidal Graphs  
Mapping onto a Circle 
Displacement, Velocity & Acceleration in Simple Harmonic Motion
 Displacement (x) = varies as the SINE function*:
 sin of time & angular frequency
 proportional to the maximum displacement
x = x_{0} sin ωt
 Velocity (v) = change in x over time = dx/dt
 Acceleration (a) = change in v over time = dv/dt
You can see that:
Component  How does it change?  Base Equation  Derived Equations 
Displacement 

x = x_{0} sin ωt


Velocity 

v = x_{0} ω cos ωt


Acceleration 

a = – x_{0} ω^{2} sin ωt

a = – ω^{2}x
Because 
All of the graphs together:
**NOTE: Phase differences
So far these graphs show examples where x = 0 when t = 0 (object begins at rest position).
But, your graph may be shifted horizontally if you start at a different position.
In other words, your phase can be different.
Imagine an oscillation which begins at maximum displacement (x_{0}).
The displacement follows another type of graph: a COSINE graph.
A cosine graph is just a sine graph with a phase difference of π/2 radians.
So now, let’s take a look at an oscillation which varies in displacement as a cosine graph.
Displacement  x = x_{0} cos ωt 
Velocity 
x = x_{0}ω sin ωt 
Acceleration 
x = x_{0}ω^{2} cos ωt 
What is a Restoring Force?
A force that acts opposite the direction of displacement in simple harmonic motion.
It is this force that causes the acceleration (a = – ω^{2}x) towards the stationary point.
The restoring force may include:
 Gravitational Force
 Force of an extended spring
How do you calculate Restoring Force?
F = ma
Since a = – ω^{2}x,
F_{res} = – mω^{2}x
Now we will look at the specifics of 2 examples: a weighted spring & a simple pendulum.
Weighted Spring
Displacement  Linear displacement: extension of spring 
Forces acting on the spring 

Restoring forces 

Motion of load 

Period of oscillation  T = 2π √(m/k)
Where

Assumptions 

Simple Pendulum
Displacement  Angular displacement: change in angle travelled by pendulum 
Forces acting on the spring 

Restoring force  Component of weight perpendicular to string = mg sin θ 
Motion of load 

Period of oscillation  T = 2π √(l/g)
Where

Assumptions 

⇐ Previous in Physics: Oscillations
⇒ Next in Physics: Energy in Oscillations