Kinetic Energy & Potential Energy of SHM (College Board AP® Physics 1: Algebra-Based): Revision Note

Kinetic energy & potential energy of SHM

Energy changes in SHM

Graph of potential and kinetic energies

• Kinetic energy and potential energy vary throughout a cycle for a system displaying SHM

• However, unlike displacement, velocity and acceleration, kinetic and potential energies only vary from zero to a maximum

  • Energy is a scalar quantity and cannot have negative values

• Where potential energy is at a maximum, kinetic energy is zero

• Where kinetic energy is at a maximum, potential energy is zero

Variation in kinetic energy

• Recall that kinetic energy is proportional to velocity squared

• At the amplitude positions:

  • Velocity is zero as the object changes direction

  • At these points, kinetic energy is zero

  • Kinetic energy cannot have a value lower than zero

• At the equilibrium position travelling in the positive direction:

  • Velocity has a maximum positive value

  • Kinetic energy has a maximum positive value

• At the equilibrium position travelling in the negative direction:

  • Velocity has a minimum negative value

  • Kinetic energy still has a maximum positive value, as squaring the velocity negates its negative sign

Variation in potential energy

• Work done by the restoring force in the opposite direction to displacement goes into the potential store

  • When the magnitudes of the restoring force and displacement are greatest, the potential energy has a maximum value

• At the equilibrium position:

  • Displacement from equilibrium is zero

  • No work is done against the restoring force

  • Potential energy of the system is zero

• At the amplitude positions:

  • The magnitudes of displacement and the restoring force are both maximum

  • Potential energy has a maximum value

Potential and kinetic energies at different positions

Total energy

• The total energy in SHM remains constant

• At every value of displacement, the sum of potential and kinetic energies is constant

  • This means when potential energy is zero (at equilibrium), maximum kinetic energy is equal to the total energy

  • Similarly, when kinetic energy is zero (at amplitude),maximum potential energy is equal to the total energy

Amplitude and total energy

• Increasing the amplitude of oscillations increases the total energy in the system

• The potential store of the SHM system is filled when the object moves against the restoring force, doing work

  • If the amplitude of oscillations is increased, more work is done so the maximum potential energy also increases

• If maximum potential energy increases, the total energy of the system increases

Total energy of an object-ideal spring system

• Recall that the elastic potential energy stored in a spring is:

$U_s=\frac{1}{2}k{\left(\mathrm{\Delta }x\right)}^2$

• Where:

  • $U_s$ = elastic potential energy stored in the spring, measured in $\mathrm{J}$

  • $k$ = spring constant measured in $\mathrm{N}/\mathrm{m}$

  • $\mathrm{\Delta }x$ = displacement, measured in $\mathrm{m}$

• When displacement is equal to amplitude, $A$, potential energy is at a maximum:

$U_{max}=\frac{1}{2}k{A}^2$

• Maximum potential energy is equal to total energy of the system:

$E_{tot}=\frac{1}{2}k{A}^2$