17.5 SHM Graphs: Displacement, Velocity, and Acceleration

In Simple Harmonic Motion, the displacement, velocity, and acceleration all vary sinusoidally with time but with specific phase relationships between them. Understanding these graphs is essential for analysing SHM.

Equations of Motion in SHM

If a particle starts from the mean position at $t=0$, its displacement is given by:

$x=x_0\sin (\omega t)$

Differentiating with respect to time gives the velocity:

$v=\frac{dx}{dt}=x_0\omega \cos (\omega t)$

This can also be written as:

$v=v_0\cos (\omega t)$

where $v_0=x_0\omega$ is the maximum (peak) velocity.

Differentiating velocity gives the acceleration:

$a=\frac{dv}{dt}=-x_0{\omega }^2\sin (\omega t)=-{\omega }^{2}x$

Graphical Representations

Displacement–Time Graph

• Shape: sine curve (if starting from mean position)
• Amplitude: $x_0$
• The curve crosses zero at $t=0,T/2,T$
• Maximum at $t=T/4$, minimum at $t=3T/4$

Velocity–Time Graph

• Shape: cosine curve
• Amplitude: $v_0=\omega x_0$
• Velocity leads displacement by $\pi /2$ radians ($90°$)
• Maximum velocity occurs at the mean position ($x=0$)
• Zero velocity at extreme positions ($x=\pm x_0$)

Acceleration–Time Graph

• Shape: negative sine curve
• Amplitude: $a_0={\omega }^2{x}_0$
• Acceleration is in anti-phase with displacement (phase difference = $\pi$ radians = $180°$)
• Maximum magnitude of acceleration at extreme positions
• Zero acceleration at the mean position

Phase Relationships Summary

Velocity as a Function of Displacement

Using the identity ${\sin }^2(\omega t)+{\cos }^2(\omega t)=1$:

$v=\pm \omega \sqrt{x_0^2-x^2}$

This formula gives the instantaneous speed at any displacement $x$:

• At $x=0$: $v=\omega x_0$ (maximum)
• At $x=\pm x_0$: $v=0$ (extreme positions)

Acceleration Using $a=-{\omega }^{2}x$

The defining equation of SHM relates acceleration directly to displacement:

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

This means:

• Acceleration is always directed towards the mean position (negative sign)
• Maximum acceleration: $a_{max}={\omega }^2{x}_0$ at $x=\pm x_0$
• Zero acceleration at $x=0$

Gradient Relationships

• The gradient of the $x$–$t$ graph at any instant equals the instantaneous velocity at that time.
• The gradient of the $v$–$t$ graph at any instant equals the instantaneous acceleration at that time.
• The steepest slope on the $x$–$t$ graph occurs at the mean position, confirming maximum velocity there.

Key Values at Special Positions