6.1.2 Simple harmonic motion (SHM)

The key equation for SHM is:

$$a = -\omega^2 x$$

Where:

• $a$ = acceleration
• $\omega$= angular speed
• $x$ = displacement from equilibrium This equation shows that as the displacement $x$ increases, the acceleration $a$ also increases but in the opposite direction, always pointing back towards the equilibrium position.

Formulas for Simple Harmonic Motion

1. Displacement as a function of time:

$$x = A \cos(\omega t)$$

where $t$ is time.

2. Velocity ($v )$ can be derived from displacement:

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

This indicates that the velocity is maximum when $x = 0$ (at the equilibrium) and zero when $x = A$ (at the amplitude).

3. Acceleration $( a )$ is derived from velocity:

$$a = -\omega^2 x$$

Here, acceleration is directly proportional to displacement but in the opposite direction.

Graphical Representation of SHM

• Displacement-Time Graph: The displacement graph is a cosine or sine curve oscillating between $+A$ and $-A$. This shows the object's back-and-forth motion with a periodic maximum displacement.

• Velocity-Time Graph: The velocity-time graph is derived as the gradient of the displacement-time graph. It oscillates between $+\omega A$ and $-\omega A$, with velocity being zero at maximum displacement points and maximum at the equilibrium position.

• Acceleration-Time Graph: The acceleration graph, derived from the gradient of the velocity-time graph, oscillates between $+\omega^2 A$ and $-\omega^2 A$. The maximum acceleration occurs when the displacement is maximum, as seen in the equation $a = -\omega^2 x$.