C.1.8 (HL)—Phase angle

Define phase angle

The phase angle $\phi$ specifies where an oscillator is within its cycle relative to a reference sine wave. It is an angular quantity measured in radians; one complete cycle is $2\pi$ radians.

Include the initial condition

For the chosen sine convention, displacement is $x=x_0\sin(\omega t+\phi)$ , and velocity is $v=\omega x_0\cos(\omega t+\phi)$ . The value of $\phi$ sets the displacement and direction of motion at $t=0$ .

Compare oscillations

A phase difference of $\pi/2$ means one oscillation is a quarter-cycle ahead of the other; $\pi$ means they are in antiphase. Choose the smallest signed or positive phase difference required by the question.

Common trap

Do not mix degrees and radians in the equations, and do not infer phase from amplitude. Phase describes timing within the cycle, not the size of the oscillation.

How it is assessed

Questions ask you to state the phase difference between two sinusoidal motions. The evidence accepts equivalent radian or degree forms, but the quarter-cycle relationship must be correct.

Command terms

State

What earns marks

Express phase in radians, identify the reference convention, and use φ to set the initial condition in x = x0 sin(ωt + φ). For two waves, compare corresponding zero crossings or peaks and report the phase difference as a fraction of a cycle, such as π/2.

Watch for

Reporting π rather than π/2 for two waves offset by a quarter cycle, or giving degrees when the question requires radians.

Representative question

[Maximum number: 1]

State the phase difference between the two waves.

«±» $\frac{\pi}{2} / 90^{\circ}$ OR $\frac{3 \pi}{2} / 270^{\circ}$

Set phase

Use radians and the phase angle to describe the initial condition: $x=x_0\sin(\omega t+\phi)$ . A phase difference of $\pi/2$ is a quarter-cycle offset.

Solve the equations

Use $v=\omega x_0\cos(\omega t+\phi)$ and $v_{\max}=\omega x_0$ . Convert $f$ to $\omega=2\pi f$ , use SI units, and retain the sign of velocity when direction is required.

Check the phase relationships

At an extreme, displacement is maximum and velocity is zero; at equilibrium, displacement is zero and speed is maximum. Displacement and velocity are one quarter-cycle out of phase.

Common trap

Do not omit $\omega$ , use degrees in a radian calculation, or treat phase angle as an amplitude.

C.1.8 (HL)—Phase angle

Model SHM with Phase Angle

C.1.8 (HL) Exam Analysis

Assessment in practice

Retrieve the HL C.1 Simple Harmonic Motion Model

Question 1