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Theme C — Wave Behaviour

C.1.2 Simple Harmonic Motion (SHM) — IB Physics HL Revision Notes & Topical Questions

5 min read 11 parts Lesson 4 of 37 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

C.1.2 Simple Harmonic Motion (SHM) — IB PHYSICS SL notes: State the conditions for SHM and describe the displacement, velocity and acceleration of an object in SHM using equations and graphs.Equilibrium — the object moves back and forth through a central position, with equal maxi…

  • Restoring force
  • Independence of amplitude
  • What the symbols mean
  • Minus sign
  • SHM systems
  • Two models

Section 1 of 11Conditions for SHM

1Equilibrium
  • Equilibrium the object moves back and forth through a central position, with equal maximum displacement on each side
2Period
  • Period each complete vibration takes the same time, so the motion is periodic
3Restoring force
  • Restoring force the force is always directed towards equilibrium and is directly proportional to the distance from it
4Independence of amplitude
  • Independence of amplitude the period does not depend on the amplitude
    • (small angles for a pendulum)
  • This is why a pendulum clock keeps time as its swing slowly dies away.

Section 2 of 11The defining equation

Simple harmonic motion
  • Simple harmonic motion oscillation in which the acceleration is proportional to the displacement and directed towards equilibrium
  • a = −ω2x
    What the symbols mean: a = acceleration (m s−2) · ω = angular frequency (rad s−1) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre
  • a is in m s−2, x in m and ω in rad s−1.
Minus sign
  • Minus sign acceleration and displacement always point in opposite directions
  • Keep the sign of displacement in calculations because it is a vector.
Restoring force
  • F = −mω2x = −kx with k = mω2
Meaning
  • Meaning a large displacement gives a large restoring force, while near the centre the force is small
  • At equilibrium the acceleration is zero but the speed is greatest, so the object overshoots.
  • The opposite force then slows it, giving regular repeating motion.

Section 3 of 11Examples and non-examples

Examples and non-examples
SHM systems
  • SHM systems a clock pendulum, a child on a swing, a mass on a spring, a vibrating guitar string, a ruler vibrating off a table edge, a tuning fork prong and molecular vibration
Two models
  • Two models you must analyse a simple pendulum and a mass–spring system, vertical or horizontal
Simplicity
  • SHM is easy to analyse and many other oscillations can be broken into a combination of SHMs
Not SHM
  • Not SHM a trampoline jumper, because the restoring force is the constant weight in the air and not proportional to displacement
  • A ball bouncing on the floor and a large-angle pendulum swing are also periodic but not SHM.
Damped systems
  • Damped systems they show decaying SHM-like motion

Section 4 of 11Behaviour through one cycle

1At an extreme
  • At an extreme the displacement is maximum, the speed is zero and the acceleration is maximum towards the centre
2Moving to the centre
  • Moving to the centre the bob accelerates towards the midpoint
3At equilibrium
  • At equilibrium the velocity is maximum and the acceleration is zero
4Past the centre
  • Past the centre the object slows towards the opposite extreme and v = 0 at x0, where it reverses

Section 5 of 11Maximum speed and acceleration

Maximum speed
  • Maximum speed vmax = ωx0, reached when passing through equilibrium
Maximum acceleration
  • Maximum acceleration amax = ω2x0, reached at the extremes
Larger amplitude
  • Larger amplitude with the same ω it gives larger speed and acceleration but no change in period
Larger ω
  • Larger ω a shorter period increases both vmax and amax

Section 6 of 11Graphs against time

1Shape
  • Shape undamped SHM graphs are sine or cosine curves, and which one depends on the starting position
2Rate of change
  • Rate of change velocity is the gradient of the x–t graph and acceleration is the gradient of the v–t graph
3Phase
  • Phase x, v and a graphs are each 90° out of phase with the next

Section 7 of 11Graphs starting at equilibrium and at the amplitude

StartDisplacementVelocityAcceleration
At equilibrium (x = 0)Sine curveCosine curveNegative sine curve
At the amplitude (x = x0)Cosine curveNegative sine curveNegative cosine curve
Displacement graph
  • Displacement graph x is maximum at the extremes and zero at equilibrium
Velocity graph
  • Velocity graph v is zero at the extremes and maximum in size at equilibrium, a quarter of a period from the x–t curve
Acceleration graph
  • Acceleration graph it is a reflection of the displacement graph in the time axis, a = −ω2x
  • When x is positive and at maximum, v = 0 and a is at its most negative.

Section 8 of 11Tests for SHM

1Acceleration–displacement graph
  • Acceleration–displacement graph a straight line through the origin with negative gradient −ω2 is the best test
2Velocity–displacement graph
  • Velocity–displacement graph it is an ellipse, traced once per cycle
3Algebra test
  • Algebra test write the net force or acceleration as a = −(constant)x and identify the constant as ω2
4Radians
  • Radians a calculator must be in radian mode for sine and cosine of ωt

Section 9 of 11More from the specification

1
Defining equation of simple harmonic motion
  • Simple harmonic motion (SHM) is characterized by an acceleration directly proportional to displacement and directed towards equilibrium: a = -omega2 x.
2
Energy exchanges in SHM
  • Total mechanical energy remains constant in an undamped harmonic oscillator, continuously exchanging between kinetic energy and potential energy.
    • (maximum at equilibrium)
3
Isochronous property of harmonic oscillators
  • The oscillation period of an SHM system is independent of amplitude, remaining constant for small angular displacements.
4
Energy interchange in SHM
  • Energy interchange in SHM vibrating mass-spring systems continuously exchange kinetic energy with elastic or gravitational potential energy, keeping total energy constant.
5
Angular frequency in oscillatory systems
  • Angular frequency in oscillatory systems angular frequency ω in radians per second relates oscillation frequency f to period via ω = 2πf.
6
Phase difference on kinematic graphs
  • Phase difference on kinematic graphs velocity-time graphs lead displacement-time curves by ninety degrees, represented by cosine and sine sinusoidal traces.
7
Kinetic energy
  • K = ½mv2, energy of motion.
8
Potential energy
  • Energy stored by the configuration of a system.
9
Gravitational potential
  • Work done per unit mass in bringing a test mass from infinity to the point.
10
Displacement-time graph
  • A graph of displacement against time whose gradient is velocity.
11
Velocity-time graph
  • A graph of velocity against time whose gradient is acceleration and area is displacement.

Section 10 of 11Key terms

TermMeaning
Simple harmonic motionMotion in which acceleration is proportional to displacement and directed to equilibrium, a = −ω²x.
Restoring forceForce that acts toward the equilibrium position.
SinusoidalHaving the shape of a sine or cosine curve.
vmaxMaximum speed, ωx₀, at equilibrium.
amaxMaximum acceleration, ω²x₀, at the extremes.

Section 11 of 11Exam technique

1
Saying acceleration is a maximum at equilibrium — it is zero there and speed is maximum.
2
Forgetting the minus sign, so the acceleration points away from equilibrium.
3
Believing the period depends on the amplitude in SHM — it does not.
4
Calling any repeated motion SHM — the condition a ∝ −x must hold.
5
Saying velocity is maximum at the extremes — it is zero where the object turns.

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Physics 0625 & 9702 SUVAT Mastery Graph Interpretation

Constant Acceleration Equations (SUVAT) & Slopes

v = u + at  |  s = ut + ½at2  |  v2 = u2 + 2as  |  s = ½(u + v)t

  • Displacement-Time Graph: Gradient = Velocity (curve = accelerating).
  • Velocity-Time Graph: Gradient = Acceleration; Area under curve = Distance travelled / Displacement.
  • Free Fall in vacuum: a = g ≈ 9.81 m/s2 downwards (air resistance negligible).
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