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SHM Derivations Step by Step (A level)

✏️ Paper first! Work out every question on paper before you open the answer. Write down every step – the equation, the numbers with units, the rearranging and the answer with its unit. In the exam, if your final answer is wrong you can still get marks for correct working, but only if the examiner can see it.

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A level Physics (AQA, OCR A and Edexcel). Work through the proof of each simple harmonic motion equation one line at a time: choose or fill in the next line, copy the key lines into your notes, then try two check questions that use the result.

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Worked solution

    Key facts

    Simple harmonic motion: the acceleration is proportional to the displacement and always directed towards the equilibrium position.

    x = A cos ωt    v = −Aω sin ωt    a = −Aω² cos ωt = −ω²x

    v = ±ω√(A² − x²)    vmax = ωA    amax = ω²A

    ω = 2π/T = 2πf

    mass–spring: T = 2π√(m/k)    simple pendulum (small angles): T = 2π√(l/g)

    Ep = ½kx²    Ek = ½mω²(A² − x²)    E = ½kA² = ½mω²A²

    Angles are in radians. In the check questions, calculated answers within 2% are marked correct.

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