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Simple Harmonic Motion

Displacement, velocity and acceleration: x = A cos ωt

  • 3 things to know
  • Unit A2 1

Overview

Recall questions on the displacement equation and the timing convention it assumes, the corresponding velocity and acceleration expressions, where in the cycle each is greatest and where zero, the maximum speed Aω and the maximum acceleration Aω², and worked calculations at a stated time.

What you need to know

Answer each one before you reveal it.

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Key term

State the equation for displacement in simple harmonic motion in the form used when timing begins at maximum displacement, and define every symbol.

Key term

Write the expression for the velocity of a body whose displacement is given by x = A cos ωt.

Key term

Write the expression for the acceleration of a body whose displacement is given by x = A cos ωt.

Questions from this subtopic

Each one with the model answer underneath.

  1. 1

    State the equation for displacement in simple harmonic motion in the form used when timing begins at maximum displacement, and define every symbol.

    Model answer

    x = A cos ωt, where x is the displacement in metres, A the amplitude in metres, ω the angular frequency in rad s⁻¹ and t the time in seconds.

  2. 2

    At what point in the cycle does the equation x = A cos ωt assume that timing begins?

    Model answer

    It assumes that timing begins when the body is at maximum displacement, because cos 0 = 1 gives x = A at t = 0.

  3. 3

    Write the expression for the velocity of a body whose displacement is given by x = A cos ωt.

    Model answer

    The velocity is v = −Aω sin ωt, in m s⁻¹, since velocity is the rate of change of displacement.

  4. 4

    Write the expression for the acceleration of a body whose displacement is given by x = A cos ωt.

    Model answer

    The acceleration is a = −Aω² cos ωt, in m s⁻², which is the same as a = −ω²x.

  5. 5

    Write the expression for the maximum speed of a body in simple harmonic motion of amplitude A.

    Model answer

    The maximum speed is ωA, in m s⁻¹, and it occurs as the body passes through the equilibrium position.

  6. 6

    Where in the oscillation does a body in simple harmonic motion move fastest, and where is it momentarily at rest?

    Model answer

    It moves fastest as it passes through the equilibrium position, and it is momentarily at rest at each extreme of the motion where the displacement is a maximum.

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