• During?simple harmonic motion, energy is constantly exchanged between two forms: kinetic and potential
• The potential energy could be in the form of:
  • Gravitational potential energy (for a pendulum)
  • Elastic potential energy (for a horizontal mass on a spring)
  • Or both (for a vertical mass on a spring)
• Speed,?v,?is at a maximum when displacement,?x, = 0, so:

The system has maximum kinetic energy when the displacement is zero because the oscillator is at its equilibrium position and so moving at maximum velocity.

• Therefore, the kinetic energy is zero at maximum displacement x = x₀, so:

• A simple harmonic system is therefore constantly converting between kinetic and potential energy
  • When one increases, the other decreases and vice versa, therefore:

The total energy of a simple harmonic system always remains constant and is equal to the sum of the kinetic and potential energies

The kinetic and potential energy of an oscillator in SHM vary periodically

• The key features of the energy-time graph are:
  • Both the kinetic and potential energies are represented by periodic functions (sine or cosine) which are varying in opposite directions to one another
  • When the potential energy is 0, the kinetic energy is at its maximum point and vice versa
  • The?total energy?is represented by a?horizontal straight line?directly above the curves at the maximum value of both the kinetic or potential energy
  • Energy is?always positive?so there are no negative values on the y axis
• Recall that the kinetic energy is defined by the equation

• Gravitational potential energy is defined by the equation

• Note:?kinetic and potential energy go through?two?complete cycles during one?period?of oscillation
  • This is because one complete oscillation reaches the maximum displacement?twice?(positive and negative)
• The energy-displacement graph for?half?a cycle looks like:

Potential and kinetic energy v displacement in half a period of an SHM oscillation

• The key features of the energy-displacement graph:
  • Displacement is a vector, so, the graph has both?positive?and?negative?x values
  • The potential energy is always at a maximum at the?amplitude?positions?x₀?and 0 at the equilibrium position (x = 0)
  • This is represented by a?‘U’ shaped curve
  • The kinetic energy is the opposite: it is 0 at the amplitude positions?x₀?and maximum at the equilibrium position x = 0
  • This is represented by a?‘n’ shaped curve
  • The total energy is represented by a?horizontal straight line?above the curves

The total energy of system undergoing?simple harmonic motion is defined by:

• Where:
  • E?= total energy of a simple harmonic system (J)
  • m?= mass of the oscillator (kg)
  • ??= angular frequency (rad s^-1)
  • x₀?= amplitude (m)

Step 1: Write down all known quantities

Mass,?m?= 23 g =?23?× 10^–3?kg

Amplitude,?x₀?= 1.5 cm =?0.015 m

Frequency,?f?=?4.8 Hz

Step 2:?Write down the equation for the total energy of SHM oscillations: