What Is Phase Angle in SHM? (HL)

Phase angle identifies the stage reached by an oscillator within one complete cycle of simple harmonic motion (SHM). In IB Physics HL, the phase angle at time $t$ is $\theta = \omega t + \phi$, measured in radians.

Because one cycle corresponds to $2\pi$ radians, the phase increases by $2\pi$ every period. Thus $\omega=2\pi/T=2\pi f$, and an elapsed-time fraction $\Delta t/T$ corresponds to a phase change $\Delta\theta=2\pi\Delta t/T$. Angles differing by $2\pi n$ describe the same stage of motion.

The Reasoning and Mechanism

For sinusoidal SHM, displacement may be written as

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

where $x_0$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase constant or initial phase angle. The complete phase angle is therefore $\theta=\omega t+\phi$.

The phase angle determines both displacement and direction because

$$x=x_0\sin\theta, \qquad v=\omega x_0\cos\theta.$$

Suppose $T=0.80\ \text{s}$, $\phi=\frac{\pi}{6}$, and $t=0.10\text{ s}$. Then

$$\omega=\frac{2\pi}{T}=2.5\pi\ \text{rad s}^{-1},$$ $$\theta=(2.5\pi)(0.10)+\frac{\pi}{6}=\frac{5\pi}{12}.$$

The oscillator is near maximum positive displacement and moving positively because $\cos\left(\frac{5\pi}{12}\right)>0$. Phase also determines acceleration: $a=-\omega^2x$. Here, , so , while acceleration points toward equilibrium.

A phase difference compares two oscillations. They are in phase at $2n\pi$ and in antiphase at $(2n+1)\pi$.

For two oscillators with the same angular frequency, $x_1=x_0\sin(\omega t+\phi_1)$ and , the phase difference is . It remains constant because the shared term cancels. If one oscillator reaches the same state later when , its phase lag is .

Phase angle is dimensionless, although radians are stated to show that it represents a cycle fraction. Do not infer direction from displacement alone: the same displacement occurs twice per cycle, but velocity's sign distinguishes the two states.

Exam Technique

This is Additional Higher Level (AHL) content in C.1. A common misconception is that phase angle means angular displacement; it does not. Identify the sine or cosine convention, calculate $\omega$, and use the signs of displacement and velocity. Keep your calculator in radians unless stated otherwise.