17.4 Phase | Simple Harmonic Motion | Physics 12

What is Phase?

Phase is the angle $θ = ω t$ that describes the complete state of an oscillating quantity at any given instant. It tells us:

The instantaneous displacement of the oscillator from its mean position.
The direction of motion (whether moving toward or away from the mean position).

Phase is measured in radians.

For a particle executing SHM, the displacement is: $x = x_{0} cos (ω t)$ Here, $ω t$ is the phase at time $t$ .

Initial Phase (Phase Constant)

In the general SHM equation: $x = x_{0} cos (ω t + φ)$ the quantity $(ω t + φ)$ is the phase at time $t$ , and $φ$ is the initial phase (or phase constant).

$φ$ is the value of the phase at $t = 0$ . It sets the starting condition of the oscillator:

Initial Phase $φ$	Starting Position
$0$	Positive extreme: $x = x_{0}$
$9 0^{\circ}$ ( $π /2$ )	Mean position: $x = 0$ , moving in negative direction
$18 0^{\circ}$ ( $π$ )	Negative extreme: $x = - x_{0}$

Phase Difference

When two oscillating quantities of the same frequency are compared, the phase difference $Δ ϕ$ is the constant angle by which one leads or lags the other.

Given: $A = A_{0} sin (ω t + ϕ_{1}), B = B_{0} sin (ω t + ϕ_{2})$

Phase difference: $Δ ϕ = ϕ_{1} - ϕ_{2}$

If $Δ ϕ > 0$ : $A$ leads $B$
If $Δ ϕ < 0$ : $A$ lags $B$
If $Δ ϕ = 0$ : $A$ and $B$ are in phase
If $Δ ϕ = 18 0^{\circ}$ : $A$ and $B$ are in antiphase (out of phase)

Phase Lead and Phase Lag

A quantity leads another if it reaches its peak value earlier in time. A quantity lags another if it reaches its peak value later in time.

Example: If $V = V_{0} cos (ω t + ϕ)$ and $I = I_{0} cos (ω t)$ , then:

Voltage leads current by angle $ϕ$
Current lags voltage by angle $ϕ$

Phasors

A phasor is a rotating vector used to represent a sinusoidal quantity graphically.

Property	Meaning
Length of phasor	Peak (amplitude) value: $V_{0}$ or $I_{0}$
Angular position	Phase angle at that instant: $ω t + φ$
Projection on reference axis	Instantaneous value of the quantity
Direction of rotation	Counter-clockwise (anti-clockwise) with angular velocity $ω$

Phasor diagrams make it easy to add two AC quantities of the same frequency by vector addition.

Phase in Purely Resistive AC Circuits

In a circuit containing only resistance, voltage and current are in phase: $ϕ = 0^{\circ}$ Both reach their maximum and minimum values at exactly the same instant.

$V = V_{0} sin (ω t), I = I_{0} sin (ω t)$

Summary of Key Terms

Term	Symbol	Definition
Displacement	$x$	Instantaneous distance from mean position
Amplitude	$x_{0}$	Maximum displacement
Period	$T$	Time for one complete oscillation
Frequency	$f$	Number of oscillations per second; $f = 1/ T$
Angular frequency	$ω$	$ω = 2 π f$ (rad s⁻¹)
Phase	$θ$	$θ = ω t$ ; specifies state of oscillator
Initial phase	$φ$	Phase at $t = 0$

What is Phase?

Phase is the angle $θ = ω t$ that describes the complete state of an oscillating quantity at any given instant. It tells us:

The instantaneous displacement of the oscillator from its mean position.
The direction of motion (whether moving toward or away from the mean position).

Phase is measured in radians.

For a particle executing SHM, the displacement is: $x = x_{0} cos (ω t)$ Here, $ω t$ is the phase at time $t$ .

Initial Phase (Phase Constant)

In the general SHM equation: $x = x_{0} cos (ω t + φ)$ the quantity $(ω t + φ)$ is the phase at time $t$ , and $φ$ is the initial phase (or phase constant).

$φ$ is the value of the phase at $t = 0$ . It sets the starting condition of the oscillator:

Initial Phase $φ$	Starting Position
$0$	Positive extreme: $x = x_{0}$
$9 0^{\circ}$ ( $π /2$ )	Mean position: $x = 0$ , moving in negative direction
$18 0^{\circ}$ ( $π$ )	Negative extreme: $x = - x_{0}$

Phase Difference

When two oscillating quantities of the same frequency are compared, the phase difference $Δ ϕ$ is the constant angle by which one leads or lags the other.

Given: $A = A_{0} sin (ω t + ϕ_{1}), B = B_{0} sin (ω t + ϕ_{2})$

Phase difference: $Δ ϕ = ϕ_{1} - ϕ_{2}$

If $Δ ϕ > 0$ : $A$ leads $B$
If $Δ ϕ < 0$ : $A$ lags $B$
If $Δ ϕ = 0$ : $A$ and $B$ are in phase
If $Δ ϕ = 18 0^{\circ}$ : $A$ and $B$ are in antiphase (out of phase)

Phase Lead and Phase Lag

A quantity leads another if it reaches its peak value earlier in time. A quantity lags another if it reaches its peak value later in time.

Example: If $V = V_{0} cos (ω t + ϕ)$ and $I = I_{0} cos (ω t)$ , then:

Voltage leads current by angle $ϕ$
Current lags voltage by angle $ϕ$

Phasors

A phasor is a rotating vector used to represent a sinusoidal quantity graphically.

Property	Meaning
Length of phasor	Peak (amplitude) value: $V_{0}$ or $I_{0}$
Angular position	Phase angle at that instant: $ω t + φ$
Projection on reference axis	Instantaneous value of the quantity
Direction of rotation	Counter-clockwise (anti-clockwise) with angular velocity $ω$

Phasor diagrams make it easy to add two AC quantities of the same frequency by vector addition.

Phase in Purely Resistive AC Circuits

In a circuit containing only resistance, voltage and current are in phase: $ϕ = 0^{\circ}$ Both reach their maximum and minimum values at exactly the same instant.

$V = V_{0} sin (ω t), I = I_{0} sin (ω t)$

Summary of Key Terms

Term	Symbol	Definition
Displacement	$x$	Instantaneous distance from mean position
Amplitude	$x_{0}$	Maximum displacement
Period	$T$	Time for one complete oscillation
Frequency	$f$	Number of oscillations per second; $f = 1/ T$
Angular frequency	$ω$	$ω = 2 π f$ (rad s⁻¹)
Phase	$θ$	$θ = ω t$ ; specifies state of oscillator
Initial phase	$φ$	Phase at $t = 0$