Phase and Phase Difference in SHM
Understanding the concept of phase, initial phase, and phase difference between two SHMs or between displacement, velocity, and acceleration.
What is Phase and Phase Difference in SHM?
The argument of the sine or cosine function in the equation of SHM, representing the state of oscillation at a given instant.
Key formula / rule: General Equation of SHM
Key points
- Define phase and initial phase in SHM.
- Calculate the phase difference between two SHMs.
- Determine the phase difference between displacement, velocity, and acceleration in SHM.
- Interpret the meaning of phase difference (in phase, out of phase).
Common exam trap
Confusing phase with angular frequency (ω) or time (t).
Definitions
- Term
Phase
- Meaning
The argument of the sine or cosine function in the equation of SHM, representing the state of oscillation at a given instant.
- Term
Initial Phase (Epoch)
- Meaning
The phase of the oscillation at time t=0.
- Term
Phase Difference
- Meaning
The difference in phase between two oscillating quantities or systems.
Learning objectives
Define phase and initial phase in SHM.
Calculate the phase difference between two SHMs.
Determine the phase difference between displacement, velocity, and acceleration in SHM.
Interpret the meaning of phase difference (in phase, out of phase).
Formulae
- Name
General Equation of SHM
- Note
A is amplitude, ω is angular frequency, t is time, φ is initial phase.
- Expression
x(t) = A sin(ωt + φ)
- Name
Velocity in SHM
- Note
Can be written as v(t) = Aω sin(ωt + φ + π/2), showing velocity leads displacement by π/2.
- Expression
v(t) = dx/dt = Aω cos(ωt + φ)
- Name
Acceleration in SHM
- Note
Can be written as a(t) = Aω² sin(ωt + φ + π), showing acceleration leads displacement by π.
- Expression
a(t) = dv/dt = -Aω² sin(ωt + φ)
- Name
Phase Difference (Two SHMs)
- Note
If ω₁ = ω₂, Δφ = φ₁ - φ₂.
- Expression
Δφ = (ω₁t + φ₁) - (ω₂t + φ₂)
Prerequisites
Understanding of circular motion.
Basic trigonometry (sine and cosine functions).
Concept of oscillation and periodic motion.
Common mistakes
Confusing phase with angular frequency (ω) or time (t).
Incorrectly calculating the phase difference between displacement and velocity/acceleration.
Assuming phase difference is always positive.
Not considering the units (radians or degrees) for phase difference.
Keywords
SHM
Phase
Initial Phase
Phase Difference
Angular Frequency
Amplitude
Displacement
Velocity
Acceleration
Oscillation
Practice preview
What is the phase difference between the displacement and velocity of a particle undergoing Simple Harmonic Motion?…
easy
Two particles are executing SHM with the same angular frequency ω. Their displacements are given by x1 = A sin(ωt) and x2 = A cos(ωt). What is the phase difference between them?…
medium
A particle executes SHM with a time period T. If it starts from its mean position at t=0, what is the phase of the particle at t = T/4?…
medium