Simple harmonic motion equations derivation

Simple harmonic motion (SHM) is a type of oscillatory motion where the acceleration of an object is always directed toward a fixed equilibrium position and is directly proportional to its displacement from that position. According to expert tutors at My Physics Buddy, mastering SHM is one of the most rewarding steps in A/AS Level Physics (9702) because it connects mechanics, calculus, and wave physics all at once.

Understanding Simple Harmonic Motion and Deriving Its Equations

The Core Idea: What Makes Motion “Simple Harmonic”?

Think about a mass hanging on a spring. When you pull it down and release it, it bobs up and down rhythmically. What governs that motion? The spring always pulls the mass back toward the rest position — and the harder you pull it away, the stronger the restoring force. That is the essence of SHM.

Formally, a body executes simple harmonic motion when its acceleration a satisfies:

a = −ω²x

where x is the displacement from the equilibrium position (in metres) and ω is the angular frequency (in rad s⁻¹). The negative sign is critical — it tells you the acceleration always opposes the displacement, pointing back toward equilibrium. This is the defining condition of SHM, and the Cambridge 9702 syllabus expects you to state it precisely in this form.

Deriving the Displacement Equation

Since a = d²x/dt², the defining condition gives us the differential equation:

d²x/dt² = −ω²x

This is a standard second-order differential equation whose general solution is:

x = A cos(ωt + φ)

where:

• A = amplitude (maximum displacement, in metres)
• ω = angular frequency (rad s⁻¹), related to period by ω = 2π/T
• t = time (seconds)
• φ = phase constant (radians), determined by initial conditions

For the common case where the object starts at maximum displacement at t = 0 (i.e., φ = 0):

x = A cos(ωt)

If instead the object starts at equilibrium and moves in the positive direction at t = 0 (φ = −π/2):

x = A sin(ωt)

Both forms are equally valid — choose whichever matches your initial conditions.

Deriving the Velocity Equation

Differentiate the displacement equation with respect to time:

v = dx/dt = −Aω sin(ωt)    (for the cosine form)

The maximum speed occurs when sin(ωt) = ±1, so:

v_max = Aω

There is also a very useful form that gives velocity as a function of displacement rather than time. Using the identity sin²(ωt) + cos²(ωt) = 1 and substituting x = A cos(ωt):

v = ±ω√(A² − x²)

This is enormously useful in exam questions where you know displacement but not time. Notice that v = 0 when x = ±A (at the turning points) and v is maximum when x = 0 (at equilibrium).

Deriving the Acceleration Equation

Differentiate velocity once more:

a = dv/dt = −Aω² cos(ωt) = −ω²x

This is exactly our defining condition — a perfect self-consistency check that confirms the solution is correct. The maximum magnitude of acceleration is:

a_max = Aω²

occurring at the turning points x = ±A, where displacement is greatest.

Summary Table of Key SHM Equations

Quantity	Equation	Maximum Value
Displacement	x = A cos(ωt)	A
Velocity	v = ±ω√(A² − x²)	Aω
Acceleration	a = −ω²x	Aω²

A Worked Example

A particle oscillates in SHM with amplitude A = 0.12 m and period T = 0.80 s. Find (a) the angular frequency, (b) the maximum speed, (c) the speed when x = 0.06 m, and (d) the maximum acceleration.

(a) Angular frequency:
ω = 2π/T = 2π/0.80 = 7.85 rad s⁻¹

(b) Maximum speed:
v_max = Aω = 0.12 × 7.85 = 0.942 m s⁻¹

(c) Speed at x = 0.06 m:
v = ω√(A² − x²) = 7.85 × √(0.12² − 0.06²) = 7.85 × √(0.0144 − 0.0036) = 7.85 × √0.0108 = 7.85 × 0.1039 = 0.816 m s⁻¹

(d) Maximum acceleration:
a_max = Aω² = 0.12 × (7.85)² = 0.12 × 61.6 = 7.39 m s⁻²

As a physics tutor with dual postgraduate degrees in Physics and Astronomy, I can tell you that students who understand where v = ±ω√(A² − x²) comes from — rather than just memorising it — answer part (c) style questions far more reliably under exam pressure. The relationship between A/AS Level Physics (9702) oscillation topics and circular motion is also worth exploring, as SHM can be elegantly visualised as the projection of uniform circular motion onto a diameter. For further reading on the mathematical foundations, the Physics Classroom SHM resource is an excellent supplement, and the Cambridge International AS & A Level Physics 9702 syllabus specifies exactly which forms of these equations you are expected to know and use.

In teaching SHM, I consistently find that the hardest leap for students is internalising the phase relationships: displacement is maximum when velocity is zero, and velocity is maximum when displacement is zero. Sketching x, v, and a all on the same time axis — even roughly — is the single most effective habit you can build for this topic. It immediately reveals that v and x are 90° out of phase, and that a and x are exactly 180° out of phase, which is just another way of seeing the minus sign in a = −ω²x.

Common Mistakes in Simple Harmonic Motion

✗ Mistake: Forgetting the negative sign in a = −ω²x and writing a = ω²x instead.
✓ Fix: Always include the minus sign — it is what defines SHM. Without it, you are describing a system that accelerates away from equilibrium, which is unstable, not oscillatory.

✗ Mistake: Using v_max = Aω when the object is not at equilibrium, applying it as if v = Aω everywhere.
✓ Fix: Use v = ±ω√(A² − x²) for velocity at any general displacement. Reserve v_max = Aω only for x = 0.

✗ Mistake: Confusing the period T with the angular frequency ω and substituting one in place of the other in equations.
✓ Fix: Always convert first using ω = 2π/T = 2πf before substituting into any SHM equation that uses ω.

Exam Relevance: Simple harmonic motion and its equations appear in Cambridge A/AS Level Physics 9702 (Paper 4), Edexcel A Level Physics, IB Physics HL, and AQA A Level Physics. All boards require both the defining condition and the displacement–velocity relationship.

Pro Tip from Koustubh B: Sketch x, v, and a versus time on one axis. Seeing all three curves together instantly locks in their phase relationships and prevents the most common SHM errors.