Simple harmonic motion

An object in simple harmonic motion (SHM) oscillates back and forth about an equilibrium position. Its acceleration always points towards equilibrium, and its magnitude is proportional to the object’s displacement from equilibrium.

The defining equation is:

a=-\omega^2x

Here, $$$a$$$ is acceleration, $$$x$$$ is displacement from equilibrium, and $\omega$ is the angular frequency, measured in radians per second. The minus sign shows that acceleration and displacement have opposite directions: on either side of equilibrium, the acceleration points back towards it.

The farther the object is from equilibrium, the greater the magnitude of its acceleration. At equilibrium, $$$x=0$$$ , so the acceleration is zero.

Displacement, velocity and acceleration in SHM

For motion starting at maximum positive displacement, the displacement, velocity and acceleration at time $$$t$$$ are:

\begin{aligned} x&=A\cos(\omega t)\\ v&=-\omega A\sin(\omega t)\\ a&=-\omega^2A\cos(\omega t) \end{aligned}

The amplitude $$$A$$$ is the maximum magnitude of displacement from equilibrium. At $$$t=0$$$ , the object is at $$$x=A$$$ , with zero velocity and acceleration towards equilibrium.

Velocity is the gradient of the displacement–time graph, and acceleration is the gradient of the velocity–time graph. All three quantities vary sinusoidally with time. Velocity is a quarter cycle ahead of displacement: for example, velocity reaches its positive maximum a quarter cycle before displacement reaches its positive maximum. Acceleration is in antiphase with displacement: when one is at its positive maximum, the other is at its negative maximum.

Speed and acceleration at different positions

Velocity can also be related directly to displacement:

v=\pm\omega\sqrt{A^2-x^2}

The sign depends on the direction of motion. The object can pass through the same displacement travelling in either direction, with the same speed but opposite velocities.

At equilibrium, displacement and acceleration are zero, while speed is greatest. At either extreme, $x=\pm A$ , velocity is zero and the magnitude of acceleration is greatest:

v_{\max}=\omega A,\qquad a_{\max}=\omega^2A