AQA A Level Physics复习笔记6.2.7 Energy in SHM

• 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 kinetic energy is at a maximum when the displacement x = 0 (equilibrium position)

• Speed v is 0 (and kinetic energy is 0) at maximum displacement x = A, so:

The potential energy is at a maximum when the displacement (both positive and negative) is at a maximum x = A (amplitude)

• 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:

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 at a 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