Simple Harmonic Motion

Simple Harmonic Motion

1. Basics

   • An object with SHM oscillates forwards and backwards either side of a midpoint
   • The distance from the midpoint is the displacement
   • There is always a restoring force that pushes the object back to the midpoint
   • The size of the restoring force depends on the displacement - the force makes the object accelerate to the midpoint
   • SHM = an oscillation in which the restoring force on an object is directly proportional to its displacement from the midpoint, and is directed towards the midpoint

2. Energies

   • The restoring force makes the object exchange KE and PE
   • The type of potential energy depends on whats providing the restoring force
   • As the object moves towards the midpoint, the restoring force does work on the object therefore PE-> KE
   • When moving away from the midpoint KE-> PE
   • At the midpoint the objects PE is 0 and its KE is max
   • At maximum displacement (amplitude) on both sides of the midpoint KE = 0 and PE = max
   • The sum of the PE and KE is called mechanical energy and is constant (as long as motion isnt damped)

3. Graphs

   • Displacement
     • Sin wave, max = A
   • Velocity
     • Phase difference of n/2 . max = wA
   • Acceleration
     • Antiphase with displacement, gradient of velocity/time graph, max = w^2 A

4. Equations

   • w = angular frequency (rad per s)
   • w = 2πf
   • f = frequency
   • w = 2π/T
   • f = 1/T
   • a = -w^2 x
   • x = displacement
   • x = Acos(wt) (WHEN T = 0 AT A MAX)
   • x = Asin(wt) (WHEN T = 0 AT THE MIDPOINT)

5. Mass on a spring

   • Investigating masses on springs requires using equations from this topic and also springs
   • F = kx
   • T = 2π √(m/k)
   • k = spring constant
   • E = 1/2 kx^2
   • When a spring stretches or compresses it stores elastic strain energy
   • F = -kx

6. Pendulum

   • A pendulum will move with SHM as long as you dont displace it too far
   • For small angles of oscillation up to 10 degrees the period of a pendulum is:
     • T = 2π √(L/g)

7. Modelling SHM

   • You can model SHM using computer programs or by hand
     • You can plot a displacement, x against time, t from an experiment
     • You can plot velocity, v against time, t from the gradient of the x/t graph at many points
     • You can plot acceleration, a against time from the gradient of v/t at many points - this graph will be the same shape as x/t but in antiphase
   • You can also model SHM with iteration

8. Free vibrations

   • Free vibrations - no transfer of energy to the surroundings
   • If no energy is transferred to and fro the surroundings then it will keep oscillating with the same amplitude forever
   • This never happens but a spring vibrating in the air is very close so is called free vibrations
   • Total energy, E = 1/2 mv^2 + 1/2 kx^2
   • or Total energy, E = KE + PE

9. Resonance

   • Resonance occurs when driving frequency = natural frequency
   • A system can be forced to vibrate by an external force
     • The vibrations it produces are called forced vibrations
     • The frequency of this force is called the driving frequency
   • When the driving frequency approaches natural frequency the system gains more and more energy and therefore vibrates with increasing amplitude - this is called resonating
   • Examples of resonance : organ pipe, swing, glass smashing by screaming and radio

10. Damping

    • Damping occurs when energy is lost to the surroundings
    • Most oscillating systems lose energy to the surroundings
    • This is due to frictional forces -called damping forces
    • Systems are often deliberately damped to stop them or minimise the effects of resonance
    • The degree of damping can vary from light damping to overdamping
    • Damping reduces the amplitude of the oscillation over time, the heavier the damping the quicker the amplitude is reduced to 0
    • Critical damping reduces the amplitude in the shortest time possible
    • Damping effects resonance aswell
      • Lightly damped systems have a very sharp resonance peak - their amplitude only increases dramatically when the driving frequency is very close to the natural frequency
      • Heavily damped systems have a flatter response - the amplitude doesnt increase much near the natural frequency and they arent as sensitive to the driving frequency

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