Simple Harmonic Motion

How to convert a displacement–time plot to x = A cos(2πt/T + φ)?

This lesson introduces the core principles of simple harmonic motion (SHM) and how it can be mathematically modeled using sine and cosine functions.

What You’ll Learn

1. Define and identify simple harmonic motion (SHM) from displacement-time graphs.
2. Differentiate between periodic motion and SHM using force–displacement relationships.
3. Calculate amplitude, time period (T), frequency (f), and angular frequency (ω).
4. Write the SHM equation using cosine or sine forms: x = A cos(ωt + φ).
5. Interpret the physical meaning of the phase constant (φ) and initial conditions.
6. Convert oscillation graphs into mathematical SHM equations.

Key Concepts Covered

• Simple harmonic motion (SHM)
• Oscillations and periodic motion
• Amplitude (A) and maximum displacement
• Frequency (f) in Hertz and Time Period (T)
• Angular frequency (ω = 2π/T)
• Phase angle / phase constant (φ)
• Displacement-time graph interpretation
• Sinusoidal wave functions (cosine/sine)

Why This Lesson is Important

Simple harmonic motion is foundational to understanding waves, oscillations, and resonance in physics. It appears in AP Physics 1, IB Physics, and engineering entrance exams like JEE and NEET. Grasping SHM allows students to model real-world systems like springs, pendulums, and alternating currents using predictable mathematical equations.

Prerequisite or Follow-Up Lessons

• Hooke’s Law and Restoring Force
• Velocity and Acceleration in SHM

Full Lesson: Simple Harmonic Motion and Sinusoidal Functions

What is Simple Harmonic Motion?

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to displacement and always directed towards the equilibrium position. Mathematically, this condition leads to sinusoidal motion.

For example, consider a particle oscillating along the x-axis, like a mass on a spring. If the mass returns to the same position and velocity repeatedly, it undergoes oscillatory motion. When this motion is governed by a restoring force ∝ displacement (F = −kx), it qualifies as SHM.

Frequency and Time Period

• Frequency (f): Number of complete oscillations per second (unit: Hertz or Hz).
• Time Period (T): Time for one full cycle; inversely related to frequency.
  T = 1/f

Example: If a particle completes 3 oscillations per second, f = 3 Hz, T = 1/3 s.

Sinusoidal Representation of SHM

The motion of a particle in SHM can be described by a sine or cosine function of time:

General Equation:
x(t) = A cos(ωt + φ)

Where:

• x(t): displacement at time t
• A: amplitude (maximum displacement)
• ω: angular frequency = 2π/T (in rad/s)
• φ: phase constant (initial phase at t = 0)

Visualizing SHM as a Graph

• At t = 0, if x = A → φ = 0, and we use cosine form.
• At t = 0, if x = 0 and particle moves left → use φ = +π/2.
• At t = 0, if x = –A → use φ = π.

Each complete oscillation traces a sinusoidal wave, with repeating cycles:

• At t = 4 s, 1 cycle completed
• At t = 8 s, 2 cycles completed
• At t = 12 s, 3 cycles completed

Writing the SHM Equation from Graph or Initial Conditions

Case 1: Starts from x = A, T = 4 s →
x = 6 cos(2πt / 4)

Case 2: Starts from x = 0, moving left →
x = 6 cos(2πt / 4 + π/2)

Case 3: Starts from x = –A →
x = 6 cos(2πt / 4 + π)

Understanding the Phase Constant (φ)

The phase constant φ:

• Determines the particle’s initial position and direction of motion.
• Changing the sign of φ changes the initial direction (left or right).
• The full phase is given by (ωt + φ), and at t = 0, the phase is just φ.

Angular Frequency and Phase

• Angular frequency ω = 2π/T = 2πf (unit: rad/s)
• It represents how quickly the particle moves through its cycle in terms of radians.

Example: For T = 4 s →
ω = 2π/4 = π/2 rad/s

Comparing Different SHM Motions

By varying A, T, and φ, we can create different SHM motions:

• Same T, different A → same frequency, different maximum displacement
• Same A, different T → same amplitude, different speeds
• Same A and T, different φ → different initial positions and directions