OCR · A-Level · Physics
Course version: H556 · Version from Sept 2015
(d) solutions to the equation a = -ω²x e.g. x = A cos(ωt) or x = A sin(ωt)
Full specification wording
Simple harmonic oscillations: (d) solutions to the equation a = -ω²x e.g. x = A cos(ωt) or x = A sin(ωt)
The defining equation of simple harmonic motion is a = -ω²x, where a is acceleration, x is displacement from equilibrium and ω is angular frequency.
Checked against the OCR H556 specification (opens in a new tab) on .
- Point 4 of 7
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
The defining equation of simple harmonic motion is a = -ω²x, where a is acceleration, x is displacement from equilibrium and ω is angular frequency. Its solutions are sinusoidal functions of time: x = A cos(ωt) or x = A sin(ωt), where A is the amplitude. Differentiating x = A cos(ωt) gives v = -Aω sin(ωt) and a = -Aω² cos(ωt), which equals -ω²x. The cosine form starts at maximum displacement when t = 0; the sine form starts at equilibrium moving in the positive direction. Both describe the same motion with a different choice of starting time, and the period is T = 2π/ω.
What earns the marks
- Recognises that x = A cos(ωt) and x = A sin(ωt) are both solutions of a = -ω²x.
- Identifies A as the amplitude and ω as the angular frequency, with ω = 2πf = 2π/T.
- Differentiates a displacement solution to obtain velocity and acceleration, showing that a = -ω²x follows.
- Distinguishes the initial conditions: cosine starts at maximum displacement, sine starts at equilibrium.
- Uses the relationship between ω, period and frequency to convert between forms.
Common mistakes and how to put them right
Treating ω as the frequency f rather than the angular frequency.
Put it right: Use ω = 2πf.
Believing that only one of the sine and cosine forms is a valid solution.
Put it right: Recognise that both satisfy the equation and differ only in initial conditions.
Dropping the minus sign in a = -ω²x, which reverses the direction of the restoring acceleration.
Put it right: Keep the negative sign because acceleration is always directed towards equilibrium.
Confusing amplitude A with displacement x.
Put it right: Note that x varies with time while A is the maximum value of x.
Exam tips
- Check the initial condition described in the question before choosing sine or cosine.
- Convert between period and angular frequency using ω = 2π/T before substituting into a solution.
- Verify a proposed solution by differentiating twice and comparing with a = -ω²x.
Try the lesson
The start of the Simple harmonic oscillations: (d) solutions to the equation a = -ω²x e.g. x = A cos(ωt) or x = A sin(ωt) lesson, made from checked teaching notes for this specification point.
Listen
The defining equation of simple harmonic motion is a = -ω²x, where a is acceleration, x is displacement from equilibrium and ω is angular frequency. Its solutions are sinusoidal functions of time: x = A cos(ωt) or x = A sin(ωt), where A is the amplitude. Differentiating x = A cos(ωt) gives v = -Aω sin(ωt) and a = -Aω² cos(ωt), which equals -ω²x. The cosine form starts at maximum displacement when t = 0; the sine form starts at equilibrium moving in the positive direction. Both describe the same motion with a different choice of starting time, and the period is T = 2π/ω. A strong answer earns marks for these points. Recognises that x = A cos(ωt) and x = A sin(ωt) are both solutions of a = -ω²x. Identifies A as the amplitude and ω as the angular frequency, with ω = 2πf = 2π/T. Differentiates a displacement solution to obtain velocity and acceleration, showing that a = -ω²x follows. Distinguishes the initial conditions: cosine starts at maximum displacement, sine starts at equilibrium. A common slip: Treating ω as the frequency f rather than the angular frequency. Instead, use ω = 2πf. Exam tip: Check the initial condition described in the question before choosing sine or cosine.
See it
Key words
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Angular frequency
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Amplitude
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Cosine form
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Sine form
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See every definition
- Angular frequency
- The quantity ω in the solutions, linked to period and frequency by ω = 2πf = 2π/T.
- Amplitude
- The quantity A in x = A cos(ωt) or x = A sin(ωt), being the maximum value of displacement x.
- Cosine form
- The solution x = A cos(ωt), which starts at maximum displacement when t = 0.
- Sine form
- The solution x = A sin(ωt), which starts at equilibrium moving in the positive direction.
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