5.6.1 Energy in SHM

• During simple harmonic motion, energy is constantly exchanged between two forms:
  • Kinetic energy
  • Potential energy

• 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)

• The speed of an oscillator is at a maximum when displacement x = 0, so:

The kinetic energy of an oscillator is at a maximum when the displacement is zero

• This is because kinetic energy is equal to  so when the oscillator moves at maximum velocity (at the equilibrium position) it reaches its maximum value of kinetic energy

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

The potential energy of an oscillator is at a maximum when the displacement (both positive and negative) is at a maximum,

• This is because the kinetic energy is transferred to potential energy as the height above the equilibrium position increases, since potential energy is equal to mgh

• 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

• The total energy is, therefore, 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

• 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 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)

• 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

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:

Step 3: Write an expression for the angular frequency

Step 4: Substitute values into energy equation

E = 2.354 × 10^–3 = 2.4 mJ