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7.4Oscillations

Energy of Simple Harmonic Oscillators — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 10 of 10 1 quiz

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Lesson summary

7.4 Energy of Simple Harmonic Oscillators — AP PHYSICS 1 notes: Describe the mechanical energy of a system exhibiting SHM.Total energy — Etotal = U + K, constant when no resistive forces do work.Systems — horizontal object–spring system has only elastic potential energy; a vertic…

  • What the symbols mean
  • Minimum kinetic energy
  • Energy against position
  • Energy against time
  • Shock absorbers
  • Energy exchange

Part 1 of 7Energy in SHM

Energy in SHM
Total energy
  • Etotal = U + K, constant when no resistive forces do work.
Systems
  • Systems horizontal object–spring system has only elastic potential energy;
    • a vertical object–spring–Earth system has elastic plus gravitational.
Elastic potential energy
  • Us = ½kx2, with x from the natural length (or from the new equilibrium if only ½kx2 is used).
Scalars
  • Scalars energies range from zero to a maximum and are never negative.

Part 2 of 7Energy values

PositionKUSpeed
x = ±A0½kA2 (all potential)0, turning point
x = 0½kA2 (all kinetic)0vmax
x = ±A/2¾E¼EBetween
x = ±A/√2½E½EEqual K and U
Total
  • E = ½kx2 + ½mv2 = ½kA2 = ½mvmax2.
    What the symbols mean: E = elastic potential energy stored (J) · k = spring constant (N m−1) · x = extension (m) · F = force (N)
Speeds
  • Speeds vmax = A√(k/m); v = √(k(A2 − x2)/m).

Part 3 of 7Dependence on amplitude

E ∝ A2
  • E ∝ A2 doubling the amplitude quadruples the energy.
Mass
  • Mass the total energy does not depend on mass.
Minimum kinetic energy
  • Minimum kinetic energy zero.

Part 4 of 7Graphs

Energy against position
  • Us is a parabola, K an inverted parabola and E a horizontal line.
Energy against time
  • K and U each oscillate at twice the frequency of the motion.

Part 5 of 7Pendulum and damping

Pendulum
  • Pendulum energy exchanges between gravitational potential mgh and kinetic energy; ½mvmax2 = mghmax.
Damping
  • Damping non-conservative forces such as air resistance transfer mechanical energy to internal energy, so the amplitude decreases.
Shock absorbers
  • Shock absorbers provide damping in cars.
Resonance
  • Resonance large amplitude when the driving frequency equals the natural frequency.

Part 6 of 7Using energy to analyse an oscillator

1
Find total energy from the amplitude: E = ½kA2.
2
At the position of interest calculate Us = ½kx2.
3
Subtract to find K = E − Us.
4
Use K = ½mv2 to find the speed.
5
Check v ≤ vmax and that energy is conserved (no damping).

Part 7 of 7More from the specification

More from the specification
Oscillatory energy invariance
  • Oscillatory energy invariance continuous conversion between kinetic and elastic potential energy provides a mathematical means to describe undamped simple harmonic oscillators.
Restoring force
  • Force directed toward equilibrium.
Displacement
  • Change in position.
Velocity
  • Rate of change of displacement.
Oscillation
  • Repeated back-and-forth motion.

Key terms

TermMeaning
Total energyE = ½kA2, constant for an ideal oscillator.
Energy exchangeTransfer between kinetic and potential energy.
DampingEnergy loss to friction that reduces amplitude.
ResonanceLarge amplitude when driving frequency equals natural frequency.
Elastic potential energyEnergy ½kx2 stored in the spring.

Avoid these mistakes

1
Using E = ½kx2 with the instantaneous x for the total energy.
2
Thinking total energy depends on the mass.
3
Saying energy is zero at equilibrium — it is all kinetic.
4
Forgetting that doubling amplitude quadruples energy.

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