How Do You Use the SHM Displacement and Velocity Equations (x = x0 sin(ωt+φ))? (HL)

For IB Physics HL (AHL), substitute the amplitude, angular frequency, time, and phase angle into $x=x_0\sin(\omega t+\phi)$. Use the corresponding velocity equation $v=\omega x_0\cos(\omega t+\phi)$, where the sign of $v$ gives the direction of motion.

The Reasoning and Mechanism

In simple harmonic motion (SHM), displacement varies sinusoidally:

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

Here, $x$ is displacement from equilibrium, $x_0$ is amplitude, $\omega$ is angular frequency in $\text{rad s}^{-1}$, is time, and is the in radians. The combined angle specifies the oscillator's position within its cycle.

Before substituting, check that all lengths use metres, time uses seconds, and the calculator is in radian mode. If frequency $f$ is given instead of angular frequency, first calculate $\omega=2\pi f$. The initial conditions determine $\phi$: evaluate the equations at $t=0$, giving $$ and .

Velocity is the rate of change of displacement. Differentiating with respect to time gives:

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

For example, suppose $x_0=0.080\text{ m}$, $\omega=4.0\text{ rad s}^{-1}$, , and . The phase is:

$$\omega t+\phi=(4.0)(0.20)+\frac{\pi}{6}=1.32\text{ rad}$$

Therefore,

$$x=(0.080)\sin(1.32)=0.077\text{ m}$$ $$v=(4.0)(0.080)\cos(1.32)=0.078\text{ m s}^{-1}$$

The result can be checked using $v=\pm\omega\sqrt{x_0^2-x^2}$. For the values above, this gives a speed of approximately , consistent with the cosine calculation after rounding. This check catches an incorrect phase or calculator mode, but the sign must still come from the direction of motion.

The positive velocity means the oscillator is moving in the positive direction. At $x=0$, speed is maximum; at $x=\pm x_0$, velocity is zero.

A common misconception is that $x_0$ means initial displacement. It is the amplitude; the initial displacement is $x(0)=x_0\sin\phi$.

Exam Technique

This is AHL content in C.1 Simple harmonic motion. For a Calculate question, show the phase calculation, substitution, signed answer, and SI unit. Keep the calculator in radian mode, and do not discard a negative velocity because it communicates direction.