# PHY C13: Simple Harmonic Motion

Today we’re covering:

• Simple Harmonic Motion
• Equations
• Graphs
• Examples of Simple Harmonic Motion
• 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 = -ω2x

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 = x0 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 Increases until reaching maximum displacement (x0) Decreases until reaching starting position Increases in opposite direction until reaching maximum displacement (-x0) Repeat cycle x = x0 sin ωt Velocity Is maximum when displacement is 0 Is 0 when displacement is maximum (x0) v = x0 ω cos ωt v0 = x0 ω v = v0 cos ωt v = ω √(x02 – x2) Acceleration Acts opposite the direction of motion Is 0 when displacement is 0 Is maximum (most negative) when displacement is maximum (x0) a = – x0 ω2 sin ωt a = – ω2x Because x0 sin ωt = x

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 (x0).
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 = x0 cos ωt Velocity x = -x0ω sin ωt Acceleration x = -x0ω2 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 = – ω2x) 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 = – ω2x,

Fres = – mω2x

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 Tension Weight of load: W = mg Restoring force of spring: F = kΔx Restoring forces Weight of load: W = mg Restoring force of spring: F = kΔx Motion of load Travels downwards due to W Restoring force causes load to decelerate until reaching x0 Load travels upwards due to F Weight causes load to decelerate until reaching -x0 Repeat cycle Period of oscillation T = 2π √(m/k) Where m = mass of load k = spring constant Assumptions Spring obeys Hooke’s Law Spring has no mass

Simple Pendulum

 Displacement Angular displacement: change in angle travelled by pendulum Forces acting on the spring Tension Weight of bob: W = mg Restoring force Component of weight perpendicular to string = mg sin θ Motion of load Travels towards equilibrium position due to mg sin θ Continues motion mg sin θ causes bob to decelerate until reaching x0 Travels back towards equilibrium position due to mg sin θ Continues motion mg sin θ causes bob to decelerate until reaching -x0 Repeat cycle Period of oscillation T = 2π √(l/g) Where l = length of pendulum g = acceleration of free fall Assumptions Bob is a point mass String has no mass & is inelastic Angular displacement is less than 5° (sin θ = θ)