Unit 7 · Topic 08

Harmonic-oscillator energy

In the ideal oscillator, kinetic and potential energy exchange without changing their sum.

Unit 7Oscillator energyOpen navigation

Concept 01

Amplitude fixes E

Essential The minimum you should retain

For a mass–spring system, U=(1/2)kx², K=(1/2)mv², and E=(1/2)kA2 E=\frac12kA^2 .

UnderstandInterpret and connect

K and U are nonnegative when elastic U is zero at equilibrium.

DeepenFormulation and conditions

Without dissipation K+U remains constant.

ExploreConnections for further study

A fixes total energy while phase fixes the instantaneous split.

Worked example

Energy and speed halfway to amplitude

m=1.0 kg, k=100 N/m, A=0.20 m, and x=0.10 m.

Given
  • m=1.0 kg
  • k=100 N/m
  • A=0.20 m
  • x=0.10 m
Target

Find E, U, K, and |v|.

  1. Total

    E=(1/2)kA2 E=\frac12kA^2 =2.0 J.

  2. Potential

    U=(1/2)kx²=0.50 J.

  3. Kinetic

    K=E−U=1.50 J.

  4. Speed

    |v|=sqrt(2K/m)=1.73 m/s.

Conclusion

E=2.0 J, U=0.50 J, K=1.50 J, and |v|≈1.73 m/s.

Mathematical relation

SHM energy

E=12kA2,U=12kx2,K=E−U,v=±ω0A2−x2 E=\tfrac12kA^2,\ U=\tfrac12kx^2,\ K=E-U,\ v=\pm\omega_0\sqrt{A^2-x^2}
Represents

Energy exchange and speed.

Physical interpretation

Energy determines |v|, not its sign.

DeepenVariables, conditions, and checks

Variables

K
kinetic energy; usual unit: J
U
elastic potential energy; usual unit: J

Conditions of application

  • Ideal system; |x|≤A.

Dimensional check

J and m/s.

normalized energy conservation

Energy fractions versus normalized position.
  • U/E
  • K/E
  • E/E

K and U are nonnegative, exchange with each other, and add to constant total energy.

Concept 02

Complete exchange

Essential The minimum you should retain

At x=±A, U=E and K=0; at x=0, U=0 and K=E.

UnderstandInterpret and connect

Speed is zero at the extrema and greatest at equilibrium.

DeepenFormulation and conditions

The energies vary at twice the frequency of x because they use squares.

ExploreConnections for further study

A constant sum on a graph checks the model.

Concept 03

Magnitude and two directions

Essential The minimum you should retain

|v|=ω₀sqrt(A²−x²) for |x|≤A.

UnderstandInterpret and connect

Energy determines magnitude, not the sign of v.

DeepenFormulation and conditions

The radicand cannot be negative for an allowed state.

ExploreConnections for further study

Choosing ± requires phase or instantaneous direction.

Concept 04

When transfer occurs

Essential The minimum you should retain

With damping, the oscillator's mechanical energy decreases.

UnderstandInterpret and connect

Energy is transformed and transferred, not destroyed.

DeepenFormulation and conditions

An external force may restore or remove energy.

ExploreConnections for further study

Steady response comes from the balance between input and dissipation.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Accepting negative K or elastic U with zero at equilibrium.

K and U are nonnegative; E is constant ideally.

Deriving one sign of v from energy alone.

Energy determines |v|; phase or direction fixes the sign.