Recognise simple harmonic motion
Use the restoring-acceleration definition of SHM and apply its acceleration equation and sinusoidal displacement solution.
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
In SHM, how is acceleration directed relative to displacement from equilibrium?
What you will learn
- Identify acceleration proportional to displacement and opposite in direction
- Use a = −ω²x
- Use x = x₀ sin ωt as a solution for SHM
A restoring acceleration defines simple harmonic motion
Simple harmonic motion occurs when acceleration is proportional to displacement from a fixed point and is always directed towards that point. With displacement x measured from equilibrium, a = −ω²x. The minus sign states that acceleration and displacement have opposite directions.
At equilibrium x = 0, acceleration is zero. At either extreme the magnitude of x, and therefore the acceleration magnitude, is greatest; acceleration points back towards equilibrium.
A sinusoid gives one solution
A solution to the acceleration equation is x = x₀ sin ωt for a suitable choice of time origin and direction. Here x₀ is amplitude and ω is angular frequency. The sine form describes repeated oscillation; a different starting phase can be represented by shifting the time origin.
Substituting a sinusoidal displacement into a = −ω²x gives acceleration proportional to the negative displacement. The displacement, acceleration and phase should be interpreted together rather than treating the minus sign as a negative magnitude.
Pause and explain
At the positive extreme of an SHM oscillation, which way does acceleration point?
Worked example
A particle in SHM has ω = 12 rad s⁻¹ and displacement x = +0.040 m. Find its acceleration and state the direction.
Show the worked solution
- Use a = −ω²x.
- Substitute ω = 12 and x = 0.040.
- Interpret the negative sign relative to the positive displacement direction.
Answer a = −12² × 0.040 = −5.76 m s⁻²; it accelerates towards equilibrium, opposite the positive displacement.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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.
An oscillator has ω = 8.0 rad s⁻¹ and x = −0.030 m. Calculate its acceleration.
Give me a hint
Use the signed displacement in a = −ω²x.
Compare my reasoning
- a = −(8.0)²(−0.030).
- a = +1.92 m s⁻².
- The positive acceleration points towards equilibrium from the negative side.
a = +1.92 m s⁻², towards equilibrium.
Look for these in your work
- Squared angular frequency
- Signed displacement substituted
- Direction interpreted
For x = x₀ sin ωt, identify the displacement at ωt = π/2 and at ωt = 3π/2.
Give me a hint
Evaluate sine at each phase angle.
Compare my reasoning
- sin(π/2) = +1, so x = +x₀.
- sin(3π/2) = −1, so x = −x₀.
- These are the two extreme positions.
The displacements are +x₀ and −x₀, respectively.
Look for these in your work
- Phase values evaluated
- Amplitude sign applied
- Extreme positions identified
At one instant an oscillator is displaced 0.020 m to the right of equilibrium. Its acceleration is 1.8 m s⁻² to the left. Find ω.
Give me a hint
Use magnitudes in |a| = ω²|x|, then take the square root.
Compare my reasoning
- ω² = 1.8/0.020 = 90 s⁻².
- ω = √90 ≈ 9.49 rad s⁻¹.
- The acceleration direction is consistent with SHM because it points opposite the displacement.
ω ≈ 9.5 rad s⁻¹.
Look for these in your work
- Magnitude relation used
- Square root taken
- Restoring direction checked
Common mistakes
- Omitting the minus sign in a = −ω²x
- Saying acceleration is constant throughout SHM
- Confusing the equilibrium point with an extreme
- Treating the chosen sine phase as the only possible starting condition
What can you explain now?
In SHM, how is acceleration directed relative to displacement from equilibrium?
Compare with the explanation
Opposite to displacement
It is a restoring acceleration towards the fixed point.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
What physical information is carried by the minus sign in a = −ω²x?
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.