(455) HL Energy equations for SHM [IB Physics HL]

Basics of Simple Harmonic Motion (SHM)

• SHM involves a mass on springs moving on a frictional surface.

• Displacement from equilibrium (

  • At x = 0, displacement is zero (equilibrium position).

  • Maximum displacement is denoted as x0.

Energy in Simple Harmonic Motion

• Potential Energy (EP):

  • At equilibrium (x = 0), potential energy is zero.

  • At maximum displacement (x0), potential energy is at its maximum.

• Kinetic Energy (EK):

  • Kinetic energy is maximum when displacement x = 0 (maximum speed).

  • At maximum displacement (x = x0), kinetic energy is zero.

  • Kinetic energy formula: EK = 1/2 mv².

Energy Graphs

• Kinetic Energy Graph:

  • Maximum at x = 0, decreases to zero at maximum displacement.

  • Parabolic shape reflecting speeds during motion.

• Potential Energy Graph:

  • Zero at x = 0, maximum at maximum displacement (x = x0).

  • Symmetric to kinetic energy graph.

• Total Energy (ET):

  • Total energy is constant, equal to the sum of kinetic and potential energy at any point.

  • Formula: ET = EP + EK.

Equations for Energy

• Potential Energy Formula:

  • EP = 1/2 * m * ω² * x²

  • m = mass (kg), ω = angular frequency (rad/s), x = displacement (m).

• Kinetic Energy derived from Potential Energy:

  • EK = 1/2 * m * (ω² * (x0² - x²)).

• Maximum Kinetic Energy:

  • EK max occurs at x = 0: EK max = 1/2 * m * ω² * x0².

• Total Energy:

  • Determined when potential energy is zero (at maximum kinetic energy): ET = 1/2 * m * ω² * x0².

Summary

• Understanding energy exchanges in SHM helps solve problems and analyze motion.

• Key equations can be derived from the fundamentals of kinetic and potential energy.