Simple Harmonic Motion (SHM) (HL IB Physics)

• Simple harmonic motion (SHM) is a specific type of oscillation where:

  • There is repetitive movement back and forth through an equilibrium, or central, position, so the maximum horizontal or vertical displacement on one side of this position is equal to the maximum horizontal or vertical displacement on the other

  • The time interval of each complete vibration is the same (periodic)

  • The force responsible for the motion (restoring force) is always directed horizontally or vertically towards the equilibrium position and is directly proportional to the distance from it

Examples of SHM

• Examples of oscillators that undergo SHM are:
  • The pendulum of a clock
  • A child on a swing
  • The vibrations of a bowl
  • A bungee jumper reaching the bottom of his fall
  • A mass on a spring
  • Guitar strings vibrating
  • A ruler vibrating off the end of a table
  • The electrons in alternating current flowing through a wire
  • The movement of a swing bridge when someone crosses
  • A marble dropped into a bowl

Examples of objects that undergo SHM

Modelling SHM

• Not all oscillations are as simple as SHM
  • This is a particularly simple kind
  • It is relatively easy to analyse mathematically
  • Many other types of oscillatory motion can be broken down into a combination of SHMs

• An oscillation is defined to be SHM when:
  • The acceleration is proportional to the horizontal or vertical displacement
  • The acceleration is in the opposite direction to the displacement (directed towards the equilibrium position)
• The time period of oscillation is independent of the amplitude of the oscillation, for small angles of oscillation
• So, for acceleration a and horizontal displacement x

a ∝ −x

• You will be required to perform calculations on and explain two models of simple harmonic motion:
  • A simple pendulum oscillating from side to side attached to a fixed point above
  • A mass-spring system oscillating vertically up and down or horizontally back and forth

Force, acceleration and displacement of a simple pendulum in SHM

Force, acceleration and displacement of a mass-spring system in SHM

An Example of not SHM

• A person jumping on a trampoline is not an example of simple harmonic motion because:
  • The restoring force on the person is not proportional to their displacement from the equilibrium position and always acts down
  • When the person is not in contact with the trampoline, the restoring force is equal to their weight, which is constant
  • This does not change, even if they jump higher

The restoring force of the person bouncing is equal to their weight and always acts downwards

• The acceleration of an object oscillating in simple harmonic motion is given by the equation:

a = −⍵²x

• Where:
  • a = acceleration (m s^-2)
  • ⍵ = angular frequency (rad s^-1)
  • x = displacement (m)

• The equation demonstrates:
  • Acceleration reaches its maximum value when the displacement is at a maximum, i.e. x = x₀ at its amplitude
  • The minus sign shows that when the object is displaced to the right, the direction of the acceleration is to the left and vice versa (a and x are always in opposite directions to each other)

• Consider the oscillation of a simple pendulum:
  • The bob accelerates as it moves towards the midpoint
  • Velocity is at a maximum when it passes through the equilibrium position 
  • The pendulum slows down as it continues towards the other extreme of oscillation
  • v = 0 at x_o as it changes direction
  • The pendulum then reverses and starts to accelerate again towards the midpoin

Graphical Representation of SHM

• The displacement, velocity and acceleration of an object in simple harmonic motion can be represented by graphs against time
• All undamped SHM graphs are represented by periodic functions
  • This means they can all be described by sine and cosine curves

• You need to know what each graph looks like and how it relates to the other graphs
• Remember that:
  • Velocity is the rate of change of displacement 
  • Acceleration is the rate of change of velocity 

Graphs that Start at the Equilibrium Position

• When oscillations start from the equilibrium position, then:
  • The displacement-time graph is a sine curve
  • The velocity-time graph is the gradient of the displacement-time graph, so a cosine graph and 90^o out of phase with the displacement-time graph
  • The acceleration-time graph is the gradient of the velocity-time graph, so a negative sine graph and 90^o out of phase with the velocity-time graph
• More information on this can be found in the IB DP Maths Differentiating Special Functions on trigonometric differentiation

The displacement, velocity and acceleration graphs in SHM are all 90° out of phase with each other

Graphs that Start at the Amplitude Position

• When oscillations start from the amplitude position, then:
  • The displacement-time graph is a cosine curve
  • The velocity-time graph is the gradient of the displacement-time graph, so a negative sine graph and 90^o out of phase with the displacement-time graph
  • The acceleration-time graph is the gradient of the velocity-time graph, so a negative cosine graph and 90^o out of phase with the velocity-time graph

The displacement, velocity and acceleration graphs in SHM are all 90° out of phase with each other

Relationship Between Graphs

• Key features of the displacement-time graphs:
  • The amplitude of oscillations A is the maximum value of x
  • The time period of oscillations T is the time taken for one full wavelength cycle

• Key features of the velocity-time graphs:
  • The velocity of an oscillator at any time can be determined from the gradient of the displacement-time graph:

  

• Key features of the acceleration-time graph:
  • The acceleration graph is a reflection of the displacement graph on the x-axis
  • This means when a mass has positive displacement (to the right), the acceleration is in the opposite direction (to the left) and vice versa (from a = −ω²x)
  • The acceleration of an oscillator at any time can be determined from the gradient of the velocity-time graph: