PHY C13: Simple Harmonic Motion

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 = -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 Image result for oscillation graph
Mapping onto a Circle Image result for oscillation graph circle gif

 


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:Screenshot 2019-05-02 at 8.34.40 PM.png

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

Screenshot 2019-05-02 at 9.38.11 PM.png

 

Velocity
  • Is maximum when displacement is 0
  • Is 0 when displacement is maximum (x0)
v = x0 ω cos ωtScreenshot 2019-05-02 at 9.38.19 PM.png

 

  • 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

Screenshot 2019-05-02 at 9.38.30 PM.png

a = – ω2x

Because
x0 sin ωt = x

All of the graphs together:
Screenshot 2019-05-02 at 9.38.39 PM.png

 

**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 ωtScreenshot 2019-05-02 at 9.44.01 PM.png
Velocity

x = -x0ω sin ωt
Screenshot 2019-05-02 at 9.44.42 PM.png

Acceleration

x = -x0ω2 cos ωt
Screenshot 2019-05-02 at 9.45.09 PM.png

 


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
  1. Travels downwards due to W
  2. Restoring force causes load to decelerate until reaching x0
  3. Load travels upwards due to F
  4. Weight causes load to decelerate until reaching -x0
  5. 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
  1. Travels towards equilibrium position due to mg sin θ
  2. Continues motion
  3. mg sin θ causes bob to decelerate until reaching x0
  4. Travels back towards equilibrium position due to mg sin θ
  5. Continues motion
  6. mg sin θ causes bob to decelerate until reaching -x0
  7. 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)

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

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