Relate velocity and displacement in SHM
Use the velocity equations and phase relationships to interpret speed and the displacement, velocity and acceleration graphs.
Section 17 · Oscillations
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 17. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use displacement, velocity, acceleration, force, energy and trigonometric functions with SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Where is the speed greatest in an SHM cycle?
What you will learn
- Use v = v₀ cos ωt for the stated sinusoidal solution
- Use v = ±ω√(x₀² − x²)
- Analyse displacement, velocity and acceleration graphs for SHM
Velocity is a quarter-cycle ahead of displacement
For x = x₀ sin ωt, velocity is v = v₀ cos ωt, where v₀ = ωx₀ is the maximum speed. At equilibrium, speed is greatest; at either extreme the velocity is zero before the object reverses direction.
The displacement graph is sinusoidal. Velocity is shifted by one quarter of a cycle: it is maximum as displacement passes through zero and zero at each displacement extreme. Acceleration is opposite to displacement, so its graph is inverted relative to x.
Position can correspond to two velocity directions
Using energy or eliminating time gives v = ±ω√(x₀² − x²). The magnitude depends on how far the object is from equilibrium; the plus or minus sign tells which direction it is moving. The same displacement is generally reached once while moving each way.
At x = 0, the speed magnitude is ωx₀. At |x| = x₀, speed is zero. A graph or position alone may not determine the sign of velocity unless the direction of travel is also given.
Pause and explain
At x = +x₀, what is the oscillator's velocity?
Worked example
An oscillator has x₀ = 0.080 m, ω = 5.0 rad s⁻¹ and x = +0.048 m. Find the possible velocities.
Show the worked solution
- Use v = ±ω√(x₀² − x²).
- Evaluate the expression inside the square root.
- Retain both signs because the object can pass this position in either direction.
Answer v = ±5.0√(0.080² − 0.048²) = ±0.32 m s⁻¹.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For an oscillator with x₀ = 0.10 m and ω = 4.0 rad s⁻¹, find its maximum speed.
Give me a hint
At equilibrium the speed is v₀ = ωx₀.
Compare my reasoning
- v₀ = ωx₀.
- v₀ = 4.0 × 0.10.
- Give the result in m s⁻¹.
v₀ = 0.40 m s⁻¹.
Look for these in your work
- Maximum-speed relation selected
- Product calculated
- Speed unit stated
A particle has x₀ = 0.050 m, ω = 10 rad s⁻¹ and x = 0.030 m. Calculate the possible velocities.
Give me a hint
Use both branches of v = ±ω√(x₀² − x²).
Compare my reasoning
- x₀² − x² = 0.0025 − 0.0009 = 0.0016 m².
- √0.0016 = 0.040 m.
- v = ±10 × 0.040 = ±0.40 m s⁻¹.
The possible velocities are +0.40 m s⁻¹ and −0.40 m s⁻¹.
Look for these in your work
- Squared positions subtracted
- Square root evaluated
- Both directions stated
Sketch in words how the acceleration–time graph compares with the displacement–time graph for SHM.
Give me a hint
Use a = −ω²x at every instant.
Compare my reasoning
- Both quantities vary sinusoidally with time.
- Acceleration is always opposite in sign to displacement.
- Their graphs are in antiphase: acceleration is inverted relative to displacement.
The acceleration graph is an inverted, scaled sinusoid compared with displacement; its magnitude scale is ω² times larger in corresponding units.
Look for these in your work
- Sinusoidal variation stated
- Opposite signs linked
- Antiphase interpretation given
Common mistakes
- Using only the positive branch without information about direction
- Claiming velocity is greatest at an extreme
- Giving displacement and acceleration graphs the same phase
- Forgetting the square root in the speed relation
What can you explain now?
Where is the speed greatest in an SHM cycle?
Compare with the explanation
At the equilibrium position
The oscillator moves fastest as it passes through x = 0.
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Make it stick
Why does knowing displacement alone usually leave two possible velocity signs?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.