Simple Harmonic Motion: Comparing Two Oscillators

Understanding Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is a special, predictable type of repeating motion. Imagine a mass on a spring or a simple pendulum swinging: they all follow SHM.

The motion is always centered around an equilibrium position (x=0), the point where the object would naturally rest. A "restoring force" (like the spring's pull) always tries to pull the object back to this center.

Here are the key terms you'll see in the simulator:

• Displacement (x): The object's position relative to the equilibrium (center) at any specific moment.
• Amplitude (A): The maximum displacement from the center. This defines the "height" of the wave and the ends of the motion.
• Period (T): The total time (in seconds) it takes to complete one full cycle of motion (e.g., from one peak, all the way to the trough, and back to the next peak).
• Phase Angle (φ): This is the "head start" of the wave, measured in radians. It's a single value that is determined by both the wave's starting position (x₀) and its initial direction of travel (its velocity, v₀).

The goal of this simulator is to plot this motion over time, creating the familiar "sine" or "cosine" wave shape.

How to Use the Simulator

This tool lets you graph and compare two different SHM waves (Wave 1 - Cyan and Wave 2 - Pink) on the same set of axes.

Main Controls

For each wave, you can adjust:

1. Amplitude (A): Changes the maximum height of the wave.
2. Start Pos (x₀): Sets the wave's exact displacement (its y-value) at time t=0. This value cannot be greater than the Amplitude.
3. Period (T): Changes the time for one full cycle. A shorter period means a faster oscillation (the wave is "scrunched" horizontally).
4. Positive initial velocity (v₀ > 0): This critical checkbox controls the wave's initial direction at t=0.

• • Unchecked (Default): The wave starts by moving down (negative velocity) or is at its positive peak (zero velocity).
  • Checked: The wave starts by moving up (positive velocity).

Live Data

• Phase Angle (φ): Located next to the "Wave 1" and "Wave 2" titles, this value is automatically calculated in radians. Watch how it changes when you adjust the Start Pos and the "Positive initial velocity" checkbox.
• Start/Stop Button: Runs or pauses the animation.
• Time (t) & Oscillations (N): The displays on the right show the current time and how many full cycles each wave has completed.

Experiments to Try (Exploring Nuances)

1. Phase Shift (cos vs. sin):

• • Set both waves to the same A and T (e.g., A=100, T=4).
  • For Wave 1, set Start Pos = 100 (its maximum Amplitude).
  • For Wave 2, set Start Pos = 0.
  • Observe the graph. Wave 1 starts at its peak (a cosine function), and Wave 2 starts at the center (a sine function). Note their different φ values.

1. The Effect of Initial Velocity:

• • Set both waves to the same A and T (e.g., A=80, T=4).
  • Set the Start Pos = 0 for both waves.
  • Leave the "Positive initial velocity" box unchecked for Wave 1.
  • Now, check the box for Wave 2.
  • Press "Start." You'll see Wave 1 starts by moving down (-sin), while Wave 2 starts by moving up (+sin).

1. Frequency vs. Period:

• • Set Wave 1 to have T = 4.
  • Set Wave 2 to have T = 2.
  • Press "Start" and watch. Wave 2 (with the shorter period) oscillates twice as fast as Wave 1.

1. Creating "Beats":

• • Set both waves to have the same A and Start Pos.
  • Give them slightly different periods (e.g., T₁ = 2.0 and T₂ = 2.1).
  • Watch how they start in sync ("in phase") but slowly drift apart, then come back together, then drift apart again.

What does the phase difference tell you about two oscillators?

In short: The phase difference Δφ = φ₁ − φ₂ tells you how far apart two oscillators are in their cycles. When the periods are equal, Δφ stays constant: Δφ = 0 means in phase (the peaks line up), Δφ = ±π means antiphase (one peaks when the other is at its lowest), and anything in between is a fixed shift. When the periods differ, Δφ keeps changing, so the oscillators drift in and out of step.

Common doubts

• “Does a bigger amplitude mean a different phase?” No. Amplitude sets how far each oscillator swings, while phase sets where it is in its cycle. The two are independent.
• “Can oscillators with different periods keep a constant phase difference?” No. If T₁ ≠ T₂, the relative phase changes continuously.
• “Which oscillator is faster, the one with the longer period?” The one with the shorter period. Frequency is f = 1 ÷ T, so f₁ ÷ f₂ = T₂ ÷ T₁. With T₁ = 4 s and T₂ = 2 s, oscillator 2 completes twice as many cycles.

Change the amplitudes, periods and starting positions of the two waves in the simulation above, press START, and watch x₁, x₂ and their sum.

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