6.2.7 Energy in SHM

• Simple harmonic motion also involves an interplay between different types of energy: potential and kinetic

  • The swinging of a pendulum is an interplay between gravitational potential energy and kinetic energy

  • The horizontal oscillation of a mass on a spring is an interplay between elastic potential energy and kinetic energy

Energy of a Horizontal Mass-Spring System

• When held so the spring is stretched beyond its equilibrium position there is the maximum amount of elastic potential energy
• When the mass is released it moves back towards the equilibrium position, accelerating as it goes causing the kinetic energy to increase
• At the equilibrium position, kinetic energy is at its maximum and elastic potential energy is at its minimum
• Once past the equilibrium position the kinetic energy decreases and elastic potential energy increases 

Energy of a Simple-Pendulum

• At the amplitude top of the swing, the pendulum has a maximum amount of gravitational potential energy
• When the pendulum is released, it moves back towards the equilibrium position, accelerating as it goes and the kinetic energy increases
• As the height of the pendulum decreases the gravitational potential energy also decreases
• Once the mass has passed the equilibrium position kinetic energy decreases and gravitational potential energy increases

Total Energy 

• The total energy remains constant, but the amount of energy in one form goes up, while the amount of energy in the other form goes down

  • This constant total energy shows how energy in a closed system is never created or destroyed; it is transferred from one store to another

  • This is the law of conservation of energy

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

• The total energy is calculated using the equation:

E = E_P + E_K

• Where:
  • E = total energy in joules (J)
  • E_P = potential energy in joules (J)
  • E_K = kinetic energy in joules (J)

Energy-Displacement Graph

• The kinetic and potential energy transfers go through two complete cycles during one period of oscillation
  • One complete oscillation reaches the maximum displacement twice (on both the positive and negative side of the equilibrium position)
• You need to be familiar with the graph showing the total, potential and kinetic energy transfers in half an SHM oscillation (half a cycle)

Graph showing the potential and kinetic energy against displacement in half a period of an SHM oscillation

• The key features of the energy-displacement graph are:
  • Displacement is a vector, so, the graph has both positive and negative x values
  • The potential energy is always maximum at the amplitude positions x = A, 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 x = A, and maximum at the equilibrium position x = 0
    • This is represented by an ‘n’ shaped curve

  • The total energy is represented by a horizontal straight line above the curves

Energy-Time Graph

• You also need to be familiar with the graph showing the total, gravitational potential and kinetic energy transfers against time for multiple cycles of a pendulum in simple harmonic motion

The kinetic and gravitational potential energy of a simple pendulum oscillating in SHM vary periodically

• The key features of the energy-time graph are:
  • Both the kinetic and potential energy transfers are represented by periodic functions (sine or cosine) which vary 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 kinetic and potential energy value
  • Energy is always positive so there are no negative values on the y axis