SHM and Uniform Circular Motion
Exploring the analogy between simple harmonic motion and the projection of uniform circular motion on a diameter.
What is SHM and Uniform Circular Motion?
An oscillatory motion in which the restoring force is directly proportional to the displacement from the equilibrium position and is directed towards the equilibrium position.
Key formula / rule: Displacement in SHM
Key points
- Understand the relationship between SHM and UCM.
- Derive the equations for displacement, velocity, and acceleration in SHM using the UCM analogy.
- Identify the parameters of SHM (amplitude, angular frequency, time period) from the parameters of UCM.
- Recognize that SHM is a specific type of oscillatory motion.
Common exam trap
Confusing angular velocity (ω) of UCM with linear velocity of SHM.
Definitions
- Term
Simple Harmonic Motion (SHM)
- Meaning
An oscillatory motion in which the restoring force is directly proportional to the displacement from the equilibrium position and is directed towards the equilibrium position.
- Term
Amplitude (A)
- Meaning
The maximum displacement of an oscillating particle from its equilibrium position.
- Term
Angular Frequency (ω)
- Meaning
A measure of how quickly an oscillation occurs, related to frequency by ω = 2πf.
- Term
Time Period (T)
- Meaning
The time taken for one complete oscillation.
- Term
Uniform Circular Motion (UCM)
- Meaning
Motion of an object in a circular path at a constant speed.
Learning objectives
Understand the relationship between SHM and UCM.
Derive the equations for displacement, velocity, and acceleration in SHM using the UCM analogy.
Identify the parameters of SHM (amplitude, angular frequency, time period) from the parameters of UCM.
Recognize that SHM is a specific type of oscillatory motion.
Formulae
- Name
Displacement in SHM
- Note
A is amplitude, ω is angular frequency, t is time, φ is initial phase.
- Expression
x(t) = A cos(ωt + φ)
- Name
Velocity in SHM
- Note
Can also be written as v = ±ω√(A² - x²).
- Expression
v(t) = -ωA sin(ωt + φ)
- Name
Acceleration in SHM
- Note
This simplifies to a = -ω²x, showing acceleration is proportional to displacement and opposite in direction.
- Expression
a(t) = -ω²A cos(ωt + φ)
- Name
Angular Frequency from UCM
- Note
Where v is the speed of the particle in UCM and A is the radius.
- Expression
ω = v/A
- Name
Time Period of SHM
- Note
The time taken for one complete oscillation.
- Expression
T = 2π/ω
Prerequisites
Uniform Circular Motion (UCM)
Basic trigonometry (sine and cosine functions)
Calculus (differentiation)
Vectors
Common mistakes
Confusing angular velocity (ω) of UCM with linear velocity of SHM.
Incorrectly relating the radius of UCM to the amplitude of SHM.
Forgetting the negative sign in the acceleration equation of SHM, which indicates the restoring nature of the force.
Assuming SHM is always horizontal or vertical without considering the projection.
Keywords
SHM
Uniform Circular Motion
Projection
Amplitude
Angular Frequency
Time Period
Restoring Force
Oscillation
Practice preview
Which of the following statements best describes the relationship between Simple Harmonic Motion (SHM) and Uniform Circular Motion (UCM)?…
easy
A particle is performing uniform circular motion in a circle of radius R. What is the nature of the motion of the projection of this particle on the diameter of the circle?…
easy
If a particle is in uniform circular motion with angular velocity ω, what is the angular frequency of the simple harmonic motion of its projection on a diameter?…
easy