Displacement, Velocity, and Acceleration in SHM
Deriving and applying mathematical expressions for displacement, velocity, and acceleration of a particle executing SHM.
What is Displacement, 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.
Key formula / rule: Displacement in SHM
Key points
- To derive the expressions for displacement, velocity, and acceleration in SHM.
- To understand the relationship between displacement, velocity, and acceleration.
- To calculate maximum and minimum values of velocity and acceleration.
- To relate these quantities to amplitude and angular frequency.
Common exam trap
Confusing angular frequency (ω) with frequency (f).
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.
- 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, measured in radians per second.
- Term
Phase Constant (φ)
- Meaning
The initial phase of the oscillation at time t=0, determining the starting position.
Learning objectives
To derive the expressions for displacement, velocity, and acceleration in SHM.
To understand the relationship between displacement, velocity, and acceleration.
To calculate maximum and minimum values of velocity and acceleration.
To relate these quantities to amplitude and angular frequency.
Formulae
- Name
Displacement in SHM
- Note
A = Amplitude, ω = Angular frequency, φ = Phase constant
- Expression
x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ)
- Name
Velocity in SHM
- Note
Maximum velocity vmax = Aω (at mean position, x=0)
- Expression
v(t) = dx/dt = Aω cos(ωt + φ) or v(t) = -Aω sin(ωt + φ)
- Name
Acceleration in SHM
- Note
Acceleration is proportional to displacement and opposite in direction: a = -ω²x. Maximum acceleration |amax| = Aω² (at extreme positions, x=±A)
- Expression
a(t) = dv/dt = -Aω² sin(ωt + φ) or a(t) = -Aω² cos(ωt + φ)
- Name
Velocity in terms of displacement
- Note
Derived from v² = (Aω cos(ωt + φ))² and x² = (A sin(ωt + φ))²
- Expression
v = ±ω√(A² - x²)
- Name
Angular frequency
- Note
f = frequency, T = time period
- Expression
ω = 2πf = 2π/T
Prerequisites
Basic trigonometry (sine and cosine functions).
Calculus (differentiation).
Concept of periodic motion.
Understanding of force and acceleration.
Common mistakes
Confusing angular frequency (ω) with frequency (f).
Incorrectly calculating derivatives of displacement to find velocity and acceleration.
Forgetting the negative sign in the acceleration equation (a = -ω²x).
Assuming velocity or acceleration is constant during SHM.
Not considering the phase constant when comparing different SHM systems.
Keywords
SHM
Simple Harmonic Motion
Displacement
Velocity
Acceleration
Amplitude
Angular Frequency
Phase Constant
Oscillation
Periodic Motion
Restoring Force
Practice preview
The displacement of a particle executing Simple Harmonic Motion (SHM) at time t, starting from its mean position, is given by:…
easy
The maximum acceleration of a particle executing Simple Harmonic Motion (SHM) with amplitude A and angular frequency omega is:…
easy
A particle executes SHM with an amplitude of 10 cm and a time period of 2 seconds. What is its maximum velocity?…
medium