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Theme C — Wave Behaviour

C.1.5 Energy Changes in Simple Harmonic Motion (SHM) — IB Physics HL Revision Notes & Topical Questions

5 min read 11 parts Lesson 7 of 37 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

C.1.5 Energy Changes in Simple Harmonic Motion (SHM) — IB PHYSICS SL notes: Describe how kinetic, potential and total energy vary with time and with displacement in SHM.Interplay — in an undamped oscillation energy passes continuously between kinetic energy EK and potential energ…

  • At the extremes
  • At equilibrium
  • In between
  • Gravitational potential energy
  • Elastic potential energy
  • Amplitude dependence

Part 1 of 9Energy exchange in SHM

Interplay
  • Interplay in an undamped oscillation energy passes continuously between kinetic energy EK and potential energy EP
  • A pendulum swaps gravitational potential and kinetic energy.
  • A horizontal mass on a spring swaps elastic potential and kinetic energy.
At the extremes
  • At the extremes the object is momentarily at rest, so all the energy is potential
At equilibrium
  • At equilibrium the speed is greatest, so EK is maximum and EP is minimum
In between
  • In between the energy is shared and EK decreases after the object passes the centre
Total energy
  • ET = EK + EP stays constant if there is no damping
  • This shows the law of conservation of energy: energy is transferred from one store to another, never created or destroyed.

Part 2 of 9Energy equations

Energy equations
Gravitational potential energy
  • EP = mgh
Elastic potential energy
  • EP = ½kx2
Kinetic energy
  • EK = ½mv2

Part 3 of 9Mass–spring system

Mass–spring system
Total energy
  • Total energy at the amplitude x0 the speed is zero, so ET = ½kx02
Amplitude dependence
  • ET is proportional to the square of the amplitude
  • Doubling the amplitude quadruples the energy.
At equilibrium
  • EK equals the total, so ½mvmax2 = ½kx02
  • This agrees with vmax = ωx0 because k = mω2.

Part 4 of 9Pendulum

Energy at the extreme
  • Energy at the extreme taking the lowest point as zero, the bob has risen by h and has EP = mgh
Energy at the bottom
  • Energy at the bottom all of that energy has become kinetic energy ½mv2
Height and angle
  • Height and angle h = l(1 − cos θ)
Pattern
  • Pattern the exchange is the same as for the spring

Part 5 of 9Energy–displacement graphs

1Potential energy
  • Potential energy a U-shaped parabola with minimum 0 at x = 0 and maximum ET at x = ±x0
2Kinetic energy
  • Kinetic energy an inverted (n-shaped) parabola with maximum ET at x = 0 and zero at x = ±x0
3Total energy
  • Total energy a horizontal straight line above the curves
4Crossing point
  • Crossing point the EK and EP curves cross at x = ±x0/√2, where each equals ET/2
5Two cycles
  • Two cycles energy transfers go through two complete cycles in one period of oscillation, because the maximum displacement is reached on both sides
6Sign
  • Sign displacement can be positive or negative but energy is never negative

Part 6 of 9Energy–time graphs

Shape
  • EK and EP are squared sinusoids that vary between 0 and ET in antiphase
Link
  • Link when EP is zero, EK is maximum and vice versa
Total
  • Total the sum is a constant horizontal line at the maximum value of either curve
Frequency
  • Frequency the energy oscillates at twice the frequency of the motion, so its period is T/2
  • EK is greatest each time the object passes the centre, in both directions.
Averages
  • Averages the mean EK and mean EP over a cycle are each ET/2

Part 7 of 9Finding speed from energy

At a given displacement
  • EK = ET − EP
Speed
  • Speed v = √(2EK/m)
Check
  • Check the answer must lie between zero and the maximum speed found from ET

Part 8 of 9Damping and energy loss

Energy loss
  • Energy loss friction and air resistance transfer energy to the thermal energy of the surroundings
  • ET decreases and the amplitude decays.
Quantitative effect
  • Quantitative effect since ET ∝ x02, a 10% fall in amplitude per cycle is about a 19% fall in energy
Graph with damping
  • Graph with damping the total-energy line falls smoothly with time, roughly exponentially, and the EK and EP peaks shrink

Part 9 of 9More from the specification

1
Restoring Force and Angular Frequency
  • In an oscillating mass-spring system, the restoring force follows Hooke's law F = -kx, generating an acceleration a = -(k/m)x;
    • comparing with SHM defining equation a = -ω²x shows that angular frequency is ω = √(k/m).
2
Kinematic Velocity and Displacement Relationships
  • The velocity of an oscillating mass at any position is calculated from v = ±ω·√(x₀² - x²);
    • maximum velocity occurs at equilibrium (x = 0),
  • while velocity drops to zero at the turning points.
3
Phase Relationships on Kinematic Graphs
  • On displacement-time, velocity-time,
  • and acceleration-time graphs of SHM, velocity leads displacement by a quarter-cycle (π/2 rad),
  • while acceleration is in antiphase (π rad) with displacement.
4
Acceleration
  • Rate of change of velocity.
5
Velocity
  • Rate of change of displacement.
6
Angular frequency
  • Ω = 2π/T = √(k/m) for a spring.
7
Restoring force
  • Force directed toward equilibrium.
8
Displacement-time graph
  • A graph of displacement against time whose gradient is velocity.
9
Angular displacement
  • Angle turned, in radians.
10
Velocity-time graph
  • A graph of velocity against time whose gradient is acceleration and area is displacement.

Key terms

TermMeaning
Kinetic energyEnergy of motion, ½mv².
Elastic potential energyEnergy stored in a stretched or compressed spring, ½kx².
Total energySum of kinetic and potential energy, constant in undamped SHM.
DampingLoss of energy from an oscillator to the surroundings.
Conservation of energyEnergy is neither created nor destroyed, only transferred.

Avoid these mistakes

1
Saying the energy is zero at the extremes — all the energy is potential there.
2
Thinking energy is lost in an ideal undamped SHM.
3
Forgetting that E_{T} is proportional to the square of the amplitude.
4
Drawing E_{K} and E_{P} against time with the same frequency as the motion — the energy frequency is double.
5
Saying E_{P} is greatest at equilibrium — it is least there.

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