Conditions for Simple Harmonic Motion (AQA A Level Physics): Revision Note

Conditions for simple harmonic motion

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

  • there is repetitive movement back and forth through an equilibrium, or central, position

  • the maximum horizontal or vertical displacement on one side the equilibrium position is equal to the maximum horizontal 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

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

$a\propto -x$

• Calculations on and explanations of two models of simple harmonic motion are required:

  • A simple pendulum oscillating from side to side attached to a fixed point above

  • Small angle approximations are applied to the simple pendulum swing, meaning that the conditions for SHM described above are valid for horizontal displacement, acceleration, and restoring force

    

  • A mass-spring system oscillating vertically up and down or horizontally back and forth

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 defining equation of SHM

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

$a=-{\omega }^{2}x$

• Where:

  • $a$ = acceleration (m s^-2)

  • $\omega$ = angular frequency (rad s^-1)

  • $x$ = displacement (m)

• The equation demonstrates:

  • acceleration reaches its maximum value when the displacement is at a maximum ie. 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 the bob pendulum again:

  • The bob speeds up as it heads 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 A as it changes direction

  • The pendulum then reverses and starts to accelerate again towards the midpoint

The graph representing a = −⍵²x

• The graph of acceleration against displacement is a straight line through the origin sloping downwards (similar to y = −x)

• The key features of the graph are:

  • the gradient is equal to −⍵²

  • the maximum and minimum displacement x values are the amplitudes −A and +A

Displacement equation

• The SHM displacement equation is:

$x=A\cos \left(\omega t\right)$

• Where:

  • $A$ = amplitude (m)

  • $\omega$ = angular frequency (rad s^-1)

  • $t$ = time (s)

• Because:

  • the graph of $x=A\cos \left(\omega t\right)$ starts from amplitude A when t = 0

  • the displacement is at its maximum when cos(⍵t) equals 1 or −1, when x = A

• The A Level revision notes on the graphs of trigonometric functions may help with understanding trigonometric graphs

• If an object is oscillating from its equilibrium position (x = 0 at t = 0) then the displacement equation will be:

$x=A\sin \left(\omega t\right)$

• The displacement will be at its maximum when sin(⍵t) equals 1 or −1, when x = A

• This is because the sine graph starts at 0, whereas the cosine graph starts at a maximum

Speed equation

• The speed of an object in simple harmonic motion varies as it oscillates back and forth

  • Where its speed is the magnitude of its velocity given by the equation:

$v=\pm \omega \sqrt{A^2-x^2}$

• Where:

  • $v$ = speed (m s^-1)

  • $A$ = amplitude (m)

  • ± = ‘plus or minus’. The value can be negative or positive

  • $\omega$ = angular frequency (rad s^-1)

  • $x$ = displacement (m)

• This comes from the fact that acceleration is the rate of change of velocity

  • When the standard equation of simple harmonic motion is differentiated using a differential equation, the above equation for velocity is obtained

• This equation shows that when an oscillator has a greater amplitude A, it has to travel a greater distance in the same time and hence has greater speed v