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SHM formulae: speed, acceleration and amplitude

Edexcel IAL Mathematics · WME03 Mechanics 3

Most SHM questions give you two facts about the motion and ask for the amplitude, period, greatest speed or greatest acceleration.

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Greatest values

From both, ω = (greatest acceleration)/(greatest speed).

Two positions and speeds

If you know the speed at two positions, write v2 = ω2(a2 - x2) twice and solve the two equations, usually by dividing one by the other to remove ω2.

Units and accuracy

ω is in rad/s. Keep exact values such as π or sqrt 5 as long as possible, and round only at the end, to 3 significant figures.

Worked example

A particle moving with SHM has speed 4 m/s when 1 m from the centre and speed 2 m/s when 2 m from the centre. Find ω, the amplitude and the period.

  1. 16 = ω2(a2 - 1) and 4 = ω2(a2 - 4).
  2. Divide: 4 = (a2 - 1)/(a2 - 4), so 4a2 - 16 = a2 - 1 and a2 = 5.
  3. ω2 = 16/4 = 4, ω = 2.
  4. T = 2π/2 = π.

Answer: ω = 2 rad/s, a = sqrt 5 = 2.24 m, T = π s

Key idea: Greatest speed aω at the centre; greatest acceleration aω2 at the ends; v2 = ω2(a2 - x2) everywhere.

Check yourself

  1. A particle moves with SHM of amplitude 0.5 m and period 2 s. Find its greatest speed, to 3 significant figures.
    Show answer

    1.57 m/s. ω = π, greatest speed = 0.5π = 1.57 m/s.

  2. For the same particle (amplitude 0.5 m, period 2 s), find its greatest acceleration, to 3 significant figures.
    Show answer

    4.93 m/s2. aω2 = 0.5π2 = 4.93 m/s2.

  3. For the same particle, find its speed when it is 0.3 m from the centre, to 3 significant figures.
    Show answer

    1.26 m/s. v2 = π2(0.25 - 0.09), v = 0.4π = 1.26 m/s.

  4. In the worked example, find the amplitude, to 3 significant figures.
    Show answer

    2.24 m. a2 = 5, a = 2.24 m.

  5. A particle moving with SHM has greatest speed 3 m/s and greatest acceleration 12 m/s2. Find the amplitude.
    Show answer

    0.75 m. ω = 12/3 = 4, so a = 3/4 = 0.75 m.

  6. A particle moving with SHM has greatest speed 3 m/s and greatest acceleration 12 m/s2. Find its period, to 3 significant figures.
    Show answer

    1.57 s. ω = 4, T = 2π/4 = 1.57 s.

  7. Where is the magnitude of the acceleration greatest in SHM?
    • At the centre
    • Halfway between the centre and an end
    • At the end points of the motion
    • It is the same everywhere
    Show answer

    At the end points of the motion. |ẍ| = ω2|x| is greatest when |x| = a.

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Last updated 8 November 2026. Spotted a mistake? Tell us.