Energy in SHM (OCR A Level Physics)

Kinetic & Potential Energies

• 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)

Energy-Displacement Graphs

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