Velocity and Acceleration in SHM
Derivation and analysis of mathematical expressions for displacement, velocity, and acceleration as functions of time in SHM, including their phase relationships and graphs.
What is Velocity and Acceleration in SHM?
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction to the displacement.
Key formula / rule: Displacement in SHM
Key points
- Derive expressions for velocity and acceleration in SHM.
- Analyze the variation of velocity and acceleration with time and displacement.
- Determine the maximum values of velocity and acceleration.
- Understand the phase relationships between displacement, velocity, and acceleration.
Common exam trap
Confusing maximum velocity with maximum acceleration.
Definitions
- Term
Simple Harmonic Motion (SHM)
- Meaning
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction to the displacement.
- Term
Amplitude (A)
- Meaning
The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
- Term
Angular Frequency (ω)
- Meaning
A measure of the rate of change of the phase angle of a sinusoidal waveform, related to frequency (f) by ω = 2πf.
- Term
Phase Angle (φ)
- Meaning
Represents the initial state of oscillation at t=0.
Learning objectives
Derive expressions for velocity and acceleration in SHM.
Analyze the variation of velocity and acceleration with time and displacement.
Determine the maximum values of velocity and acceleration.
Understand the phase relationships between displacement, velocity, and acceleration.
Interpret graphical representations of displacement, velocity, and acceleration vs. time.
Formulae
- Name
Displacement in SHM
- Note
A = Amplitude, ω = Angular frequency, t = time, φ = initial phase.
- Expression
x(t) = A sin(ωt + φ)
- Name
Velocity in SHM
- Note
Maximum velocity vmax = Aω occurs at x=0.
- Expression
v(t) = dx/dt = Aω cos(ωt + φ)
- Name
Velocity in terms of displacement
- Note
Derived from v(t) and x(t) using sin²θ + cos²θ = 1.
- Expression
v = ±ω√(A² - x²)
- Name
Acceleration in SHM
- Note
Maximum acceleration amax = Aω² occurs at x=±A.
- Expression
a(t) = dv/dt = -Aω² sin(ωt + φ)
- Name
Acceleration in terms of displacement
- Note
This is the defining equation for SHM, showing acceleration is proportional to displacement and opposite in direction.
- Expression
a = -ω²x
Prerequisites
Understanding of periodic motion.
Basic calculus (differentiation).
Trigonometric functions (sine and cosine).
Concept of amplitude, angular frequency, and phase.
Definition of velocity and acceleration.
Common mistakes
Confusing maximum velocity with maximum acceleration.
Forgetting the negative sign in the acceleration formula (a = -ω²x), which indicates direction.
Incorrectly assuming velocity or acceleration to be constant.
Errors in calculating phase differences.
Using vmax = Aω or amax = Aω² at the wrong positions (e.g., vmax at extremes).
Keywords
SHM
Simple Harmonic Motion
Velocity
Acceleration
Amplitude
Angular Frequency
Phase
Oscillation
Periodic Motion
Practice preview
What is the phase difference between displacement and velocity in Simple Harmonic Motion (SHM)?…
easy
A particle executes SHM with a displacement given by x(t) = 5 sin(4πt) meters. What is its maximum speed?…
medium
What is the acceleration of a particle executing Simple Harmonic Motion (SHM) at its mean position?…
easy