Phase Angles in Simple Harmonic Motion (SHM) (HL) | HL IB Physics Revision Notes 2025

Phase Angles in Simple Harmonic Motion

Two points on a sine wave, or on different waves, are in phase when they are at the same point in their wave cycle
- The angle between their wave cycles is known as the phase angle

3-1-7-phase-angles-circle-sine-wave-comparison

The relationship between a sine wave and phase angle

If an oscillation does not start from the equilibrium position, then it will be out of phase by an angle of $ϕ$
- This would be compared to an oscillation which does start from the equilibrium position

The phase angle $ϕ$ of an oscillation (in SHM) is defined as

The difference in angular displacement compared to an oscillator which has a displacement of zero initially (i.e. $x = 0$ when $t = 0$ )

The phase angle can vary anywhere from 0 to 2π radians, i.e. one complete cycle
With the inclusion of the phase angle $ϕ$ , the displacement, velocity and acceleration SHM equations become:

$x = x_{0} \sin (ω t + ϕ)$

$v = ω x_{0} \cos (ω t + ϕ)$

$a = - ω^{2} x_{0} \sin (ω t + ϕ)$

If two bodies in simple harmonic motion oscillate with the same frequency and amplitude, but are out of phase by $\frac{π}{2}$ , then:
- The displacement of the oscillator starting from the equilibrium position is represented by the equation $x = x_{0} \sin ω t$

- The displacement of the oscillator which leads by $\frac{π}{2}$ is represented by the equation $x = x_{0} \sin (ω t - \frac{π}{2})$

3-1-7-phase-angle-shm-example

Two oscillators which are out of phase by $ϕ = \frac{π}{2}$ . The blue-dotted wave represents an oscillator starting from the equilibrium position and the red wave represents an oscillator leading by $\frac{π}{2}$

When a sine wave leads by a phase angle of $\frac{π}{2}$ , this is equivalent to the cosine of the wave

$x = x_{0} \sin (ω t - \frac{π}{2}) = x_{0} \cos ω t$

Alternatively, a sine wave can be described as a cosine wave that lags by $\frac{π}{2}$

$x = x_{0} \cos (ω t + \frac{π}{2}) = x_{0} \sin ω t$

Notice:
- For a wave that lags the phase difference is $+ \frac{π}{2}$
- For a wave that leads the phase difference is $- \frac{π}{2}$
This is the opposite sign to the one you might think.
- To review this concept, use the notes on the Transformation of Trigonometric Functions

How are sine and cosine functions related?

3-1-7-phase-angle-sine-cosine-relationship3-1-7-phase-angle-sine-cosine-relationship

Sine and cosine functions are simply out of phase by $\frac{π}{2}$ radians

The general rules for phase shifts of sine and cosine functions are shown in the table below

Graph	Equation	Phase shift
	$\sin (ω t - ϕ)$	Shifts by $ϕ$ to the right (positive direction)
	$\sin (ω t + ϕ)$	Shifts by $ϕ$ to the left (negative direction)
	$\cos (ω t - ϕ)$	Shifts by $ϕ$ to the right (positive direction)
	$\cos (ω t + ϕ)$	Shifts by $ϕ$ to the left (negative direction)

Worked example

An object oscillates with simple harmonic motion which can be described by the equation

$x = x_{0} \cos (ω t - \frac{π}{2})$

Which of the following graphs correctly represents this equation?

3-1-7-phase-angle-shm-mcq-worked-example

Answer: C

The equation $x = x_{0} \cos (ω t - \frac{π}{2})$ is equivalent to $x = x_{0} \sin (ω t)$
This describes an oscillation where $x = 0$ when $t = 0$
- Hence, options A & D are not correct

When $\sin (ω t)$ is positive, the oscillation will start moving in the $+ x$ direction
- Hence, option B is not correct

Worked example

A mass attached to a spring is released from a vertical height of $h_{m a x}$ at time $t = 0$ . The mass oscillates with a simple harmonic motion of period T.

3-1-7-phase-angle-shm-mass-spring-worked-example1

The graph shows the variation of h with t.

3-1-7-phase-angle-shm-mass-spring-worked-example2

(a)

State the equation of motion for this oscillation.

(b)

A second mass-spring system is set up and made to oscillate with the same frequency but with a phase angle of

ϕ = - \frac{π}{4}

. On the graph, sketch the variation of h with t for the second mass-spring system.

Answer:

(a)

The displacement-time equation for an oscillator released from a maximum displacement has the form

$x = x_{0} \cos ω t$

$x = x_{0} \sin (ω t - \frac{π}{2})$

As the graph is leading a normal sine graph by $\frac{π}{2}$

Where $x = h$ and $x_{0} = h_{m a x}$
Angular frequency $ω$ is equal to

$ω = \frac{2 π}{T}$

Therefore, the equation of motion for this oscillation is:

$h = h_{m a x} \cos (\frac{2 π}{T} t)$

$h = h_{m a x} \sin (\frac{2 π}{T} t - \frac{π}{2})$

(b)

One complete oscillation is equivalent to 2π rad

SHaGKqtx_3-1-7-phase-angle-shm-mass-spring-worked-example-ma

A phase angle of $ϕ = - \frac{π}{4}$ corresponds to a shift to the right (positive direction)

Phase Angles in Simple Harmonic Motion (SHM) (HL) (HL IB Physics)

Revision Note