Using the Simple Harmonic Motion (SHM) Simulator: A Student Guide

This interactive simulation helps you visualize Simple Harmonic Motion (SHM) and connect the mathematical equation x(t) = A·cos(ωt + φ) to a physical system (the oscillating mass) and its graphical representation (the wave).

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How to Use This Simulation

1. Start with One Wave
   • Begin with Wave 1 only. Get comfortable with how each control changes the motion and the graph.
2. Adjust Amplitude (A)
   • Change the Amplitude (A). It is the “maximum displacement.”
   • As A increases, the block travels farther from the center and the wave on the graph becomes taller.
   • Changing A does not affect how long one cycle takes.
3. Adjust Period (T)
   • Increase T to make each oscillation take more time (motion looks slower, graph stretches horizontally).
   • Decrease T to make oscillations faster (graph compresses horizontally).
4. Set Start Position (x₀)
   • Try x₀ = 0, 50, 100 (with A in the same units).
   • x₀ is the position of the mass at time t = 0.
5. Use the “Positive Initial Velocity” Checkbox
   • This toggle sets the initial direction of motion at t = 0.
   • Example: with x₀ = 50 (and A = 100), checking the box makes the mass initially move away from the center (positive velocity); unchecking makes it head toward the center (negative velocity).
   • Notice how this flips the early shape of the graph.
6. Compare Two Waves
   • Turn on Wave 2 and choose different parameters.
   • Try making T₂ = ½·T₁. Observe how many oscillations Wave 2 completes while Wave 1 completes one.
   • Set both waves to the same A and T but check “Positive Initial Velocity” for only one of them. Notice how their graphs become mirror images initially.

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Key Concepts & Nuances to Notice

1. The “Live” Equation
   • The equation x = A·cos(2πt/T + φ) shown under the mass updates instantly as you adjust the controls.
   • This is the direct link from your chosen parameters (A, T, x₀ and initial velocity) to the final mathematical formula.
2. Amplitude vs. Period
   • Changing Amplitude (A) does not change Period (T).
   • A larger A means the mass travels farther but also moves faster, completing a cycle in the same time. This is a core property of SHM (for ideal springs and small angles in pendulums).
3. Understanding Phase Shift (φ)
   • The simulator computes φ from your Start Pos (x₀) and the initial velocity choice.
   • Useful anchor cases:
   – Start Pos x₀ = A (e.g., A = 100) ⇒ φ = 0. This is the standard cos graph starting at its maximum with v = 0.
   – Start Pos x₀ = 0 with negative initial velocity (box unchecked) ⇒ φ ≈ +1.57 (π/2). This corresponds to a sin graph.
   – Start Pos x₀ = 0 with positive initial velocity (box checked) ⇒ φ ≈ −1.57 (−π/2). This corresponds to a −sin graph.
4. The “Start Pos” Clamp
   • If you set Start Pos beyond the Amplitude (e.g., A = 50, Start Pos = 80), the value clamps back to 50.
   • Reason: the system can’t start at a position it cannot physically reach.
5. Velocity Checkbox at the Extremes
   • If Start Pos equals Amplitude (e.g., A = 100, x₀ = 100), toggling “Positive initial velocity” has no effect on φ.
   • At a maximum (or minimum) position, the mass is momentarily at rest (v = 0) before turning around, so you can’t assign a positive or negative initial velocity there.

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Tips for Study and Practice

• Predict first: Before pressing Start, predict how the graph will look after you change A, T, x₀, or the velocity checkbox—then run and check your intuition.
• Match motion and graph: Track where the block is (left/right of center, moving toward/away) and match that to whether the graph is above/below the axis and rising/falling.
• Build from simple to complex: Master Wave 1; then use Wave 2 to compare frequency, phase, and starting direction differences side-by-side.

Using the SHM simulator for wave comparison

This simulation helps you visualize Simple Harmonic Motion (SHM). Use it to connect the equation x(t) = A·cos(ωt + φ) to an oscillating mass and its graph.

SHM Wave Simulator Guide

This tool compares two Simple Harmonic Motion (SHM) waves (Wave 1 - Cyan, Wave 2 - Pink) on one graph.

Controls

• Amplitude (A): Sets the wave's maximum height.
• Start Pos (x₀): The wave's position at t=0.
• Period (T): Time (in seconds) for one full oscillation.
• Positive initial velocity (v₀ > 0): Check this box to make the wave start by moving up (positive velocity). If unchecked, it starts moving down or from a peak.
• Phase Angle (φ): (Next to the title) This is auto-calculated in radians based on your Start Pos and initial velocity.
• Shared Graph Duration: Sets the total time on the x-axis.
• Start/Stop: Runs or pauses the animation.

Things to Try!

1. Phase Shift: Set both waves to the same A and T. Set x₀ for Wave 1 to its max (A) and x₀ for Wave 2 to 0. Observe how Wave 1 starts at its peak (cos) and Wave 2 starts at the center (sin).
2. Initial Velocity: Set both waves to the same A and T, and set x₀=0 for both. Check the "Positive initial velocity" box for Wave 2 only. Watch Wave 1 start by moving down while Wave 2 moves up.
3. Frequency: Set T=4 for Wave 1 and T=2 for Wave 2. Watch Wave 2 oscillate twice as fast.
4. Beats: Set both waves to identical settings, but give them slightly different periods (e.g., T₁=2.0, T₂=2.1). Watch them drift in and out of phase.

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