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Theme C — Wave Behaviour

C.1.6 Equations for Simple Harmonic Motion (SHM) — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 8 of 37 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

C.1.6 Equations for Simple Harmonic Motion (SHM) — IB PHYSICS HL notes: Use the sine and cosine equations for displacement, velocity and acceleration in SHM and the relation v = ±ω√(x0² − x²).Projection — SHM is the projection of uniform circular motion onto a diameterSame quanti…

  • Same quantities
  • Choice of function
  • At x = ±x0
  • Amplitude link
  • Maximum speed
  • Maximum acceleration

Part 1 of 7Link to circular motion

Projection
  • SHM is the projection of uniform circular motion onto a diameter
Same quantities
  • Same quantities the circle of radius x0 and angular speed ω gives a sinusoid with the same amplitude and period
Radians
  • Radians the angle ωt is in radians, so ω = 2π/T appears in the equations

Part 2 of 7Starting from equilibrium (x = 0 at t = 0)

Displacement
  • Displacement x = x0 sin ωt
Velocity
  • Velocity v = ωx0 cos ωt, the derivative of x
Acceleration
  • Acceleration a = −ω2x0 sin ωt = −ω2x

Part 3 of 7Starting from the amplitude position

Starting from the amplitude position
Displacement
  • Displacement x = x0 cos ωt
Velocity
  • Velocity v = −ωx0 sin ωt
Acceleration
  • Acceleration a = −ω2x0 cos ωt
Choice of function
  • Choice of function sine or cosine depends only on the starting position, the motion is the same SHM
Symbols
  • Symbols x in m, x0 amplitude in m, ω in rad s−1, t in s

Part 4 of 7Velocity–displacement relation

Equation
  • Equation v = ±ω√(x02 − x2)
  • It follows from integrating the defining equation of SHM, or from sin2 + cos2 = 1
At x = ±x0
  • At x = ±x0 the speed is zero
At x = 0
  • At x = 0 the speed is a maximum
Sign
  • Sign plus or minus gives the two directions of motion
Amplitude link
  • Amplitude link a larger amplitude means a greater distance in the same period so a greater speed

Part 5 of 7Maximum values

1Maximum speed
  • Maximum speed vmax = ωx0 at equilibrium
2Maximum acceleration
  • Maximum acceleration amax = ω2x0 at the extremes
3Phase
  • Phase when x is at a peak, v is zero and a is at a negative peak

Part 6 of 7Graphs of the equations

Graphs of the equations
Displacement graph
  • Displacement graph a sine or cosine curve of amplitude x0 and period 2π/ω
Velocity graph
  • Velocity graph amplitude ωx0, a cosine curve if x is a sine curve, and ahead of x by a quarter cycle
Acceleration graph
  • Acceleration graph amplitude ω2x0 and an inverted copy of the x graph
Gradients
  • Gradients the gradient of x–t gives v and the gradient of v–t gives a
Velocity against displacement
  • Velocity against displacement an ellipse with semi-axes x0 and ωx0

Part 7 of 7More from the specification

1Kinematic equations for SHM
  • Displacement and velocity of an oscillator vary sinusoidally with time: x = x0 cos(omega t) and v = -omega x0 sin(omega t).
2Velocity-displacement relationship
  • Velocity at any displacement x is given by v = +-omega sqrt(x02 - x2, reaching maximum magnitude vmax = omega x0 at equilibrium.
3Period equations for pendulums and springs
  • The period of a simple pendulum is T = 2pi sqrt(L/g), while a mass-spring system obeys T = 2pi sqrt(m/k).
4Energy conservation in simple pendulums
  • Energy conservation in simple pendulums during pendulum oscillation, gravitational potential energy transforms fully into kinetic energy as the bob passes equilibrium.
5Natural frequency of mass-spring oscillator
  • Natural frequency of mass-spring oscillator the natural frequency of an oscillating mass-spring system depends solely on mass m and spring stiffness k.
6Kinematic equations for harmonic oscillators
  • Kinematic equations for harmonic oscillators displacement of an oscillating object varies sinusoidally with time, reaching zero velocity at extreme amplitude displacements.
7Potential energy
  • Energy stored by the configuration of a system.
8Kinetic energy
  • K = ½mv2, energy of motion.
9Restoring force
  • Force directed toward equilibrium.
10Gravitational potential
  • Work done per unit mass in bringing a test mass from infinity to the point.
11Displacement-time graph
  • A graph of displacement against time whose gradient is velocity.
12Velocity-time graph
  • A graph of velocity against time whose gradient is acceleration and area is displacement.
13Oscillation
  • Repeated back-and-forth motion.
14Frequency
  • Revolutions per second, f = 1/T.
    What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)
15Pendulum
  • A weight swinging on a string, exchanging GPE and KE.
16Constant
  • Not changing.

Key terms

TermMeaning
x = x0 sin ωtDisplacement of SHM starting at equilibrium.
x = x0 cos ωtDisplacement of SHM starting at an extreme.
v = ±ω√(x02 − x2)Speed as a function of position.
Maximum speedωx0, reached at equilibrium.
Maximum accelerationω2x0, reached at the extremes.

Avoid these mistakes

Using a calculator set to degrees — ωt must be in radians.
Using sine when the object starts at the extreme position — that case needs cosine.
Forgetting the ± sign in the v–x equation — it gives the direction of motion.
What the symbols mean:
Mixing up ω and f — ω = 2πf.
What the symbols mean: ω = angular frequency (rad s−1) · f = frequency (Hz) · T = period (s)
Writing vmax = x0/ω — the correct form is ωx0.

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