SHM Equations

Now that we have established the defining equation of SHM (a = −ω²x), we need to develop the full set of equations that describe how displacement, velocity, and acceleration vary with time. These equations are the tools you will use to solve virtually every SHM problem at A-Level.

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Displacement as a Function of Time

The displacement of an object undergoing SHM varies sinusoidally with time. Depending on the starting conditions, we use either:

x = A cos(ωt) (if the object starts at maximum displacement)

x = A sin(ωt) (if the object starts at the equilibrium position moving in the positive direction)

where:

• x is the displacement at time t (m)
• A is the amplitude — the maximum displacement (m)
• ω is the angular frequency (rad s⁻¹)
• t is the time elapsed from the starting point (s)