Simple harmonic motion

  • Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

  • Key elements in the study of SHM include amplitude, frequency, period, phase difference and displacement.

  • Amplitude is the maximum displacement from the equilibrium position. It is always a positive quantity.

  • Frequency refers to how often the oscillations occur. It is measured in hertz (Hz) and is the reciprocal of the period.

  • Period refers to the time it takes for one complete cycle of the oscillation.

  • Phase difference defines the position of a particle in its period of motion at t = 0.

  • The equations of SHM can be modelled in the form, a = - (ω^2) x or x = A cos (ωt + α), where
    • ‘a’ is acceleration,
    • ‘x’ is displacement,
    • ‘A’ is amplitude of motion,
    • ‘ω’ (omega) is the angular frequency of motion (2π times frequency in Hz), and
    • ‘α’ (alpha) is the phase constant that describes the initial conditions of motion at t=0.

  • Key components of SHM graphs include sinusoidal displacements and waveforms.

  • Mathematical operations on SHM such as differentiation and integration involve common trigonometric functions such as sine and cosine.

  • Energy in a system undergoing SHM is constant, if we discount friction. The total energy consists of the kinetic energy (1/2 mv^2) and potential energy (1/2 kA^2), where m is mass, v is velocity, k is the constant of proportionality from Hooke’s law and A is the amplitude of motion.

  • SHM bears close relation to circular motion and can be represented as a projection of circular motion.

  • Damping is present in SHM systems and it reduces energy in the system- this can take the forms of light, heavy or critical damping.

  • The moodelling of SHM often involves understanding and interpretation of real-life contexts, such as a swinging pendulum or a buoy bobbing on waves.