Energy in SHM (AQA A Level Physics): Revision Note

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

• The system has the maximum amount of elastic potential energy when held so the string is stretched beyond its equilibrium position

• 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 at the 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 of an SHM system

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

• Remember the equations for potential and kinetic energy:

  • Gravitational potential energy: $E_p=mgh$

  • Elastic potential energy, $E_p=\frac{1}{2}k{x}^2$

  • Kinetic energy, $E_k=\frac{1}{2}m{v}^2$

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)

• Familiarity with the graph showing the total, potential and kinetic energy transfers in half an SHM oscillation (half a cycle) is required

• 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 for a simple pendulum

• Familiarity with the graph showing the total, gravitational potential and kinetic energy transfers against time for multiple cycles of a simple pendulum oscillating in simple harmonic motion is also required

• The key features of the simple pendulum energy-time graph are:

  • both the kinetic and gravitational potential energy transfers are represented by periodic functions (sine or cosine) which vary in opposite directions to one another

  • when the gravitational potential energy is 0, the kinetic energy is at its maximum and vice versa

  • the total energy is represented by a horizontal straight line directly above the energy curves at the maximum kinetic and gravitational potential energy value

  • energy is always positive so there are no negative values on the y-axis (Any SHM energy graph drawn with negative energy values is incorrect)