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Document Brief: Constant Velocity Antiphase Model Using Tracker

This document explores the modeling of two oscillatory objects moving at constant velocities in antiphase using Tracker software. The system simulates antiphase behavior where the two objects maintain opposite phases (180∘ out of phase) as they oscillate, with their motions synchronized but inverted.

Purpose:

To investigate the behavior of two oscillating objects moving at constant velocity in antiphase and analyze the displacement, velocity, and phase relationships between them.

Key Features:

• Tracking and analysis of two synchronized oscillatory motions.
• Examination of antiphase behavior (ϕ=π phase difference).
• Graphical representation of displacement, velocity, and phase differences.

Study Guide: Modeling Antiphase Oscillations with Constant Velocity

Learning Objectives:

1. Understand the concept of antiphase motion in oscillatory systems.
2. Analyze displacement and velocity graphs to identify antiphase behavior.
3. Validate the phase relationship and motion dynamics between the two objects.

Step-by-Step Guide:

1. Setup and Calibration:

   • Import the video of the two oscillating objects into Tracker.
   • Calibrate the scale using a visible reference for accurate motion measurements.

2. Tracking Motion:

   • Track the positions of both objects frame by frame.
   • Record displacement and velocity data for both objects over multiple oscillations.

3. Define Position Functions:

   • Define the position functions for the objects:
     • Object 1: x1(t)=Asin⁡(ωt+ϕ1)
     • Object 2: x2(t)=Asin⁡(ωt+ϕ2),
     • where ϕ1=0 and ϕ2=π for antiphase motion.

4. Graphical Analysis:

   • Plot displacement (x1,x2) vs. time (t):
     • Verify the sinusoidal motion of both objects.
     • Observe that the objects are 180∘ out of phase.
   • Plot velocity (v1,v2) vs. time:
     • Confirm that the velocity graphs also reflect antiphase behavior.

5. Applications:

   • Study wave interference patterns (e.g., destructive interference in ripple tanks).
   • Explore coupled oscillatory systems with opposite phase relationships.

Tips for Success:

• Ensure accurate tracking of both objects to validate the phase difference.
• Use consistent scaling for displacement and velocity graphs to highlight antiphase relationships.

FAQ: Constant Velocity Antiphase Model

1. What does antiphase motion mean?

Antiphase motion occurs when two oscillatory objects have a phase difference of $\pi$ radians (180∘), meaning one object reaches its peak while the other reaches its trough.

2. How is the phase difference maintained?

In this model, the phase difference is constant due to synchronized oscillations. The displacement functions for the two objects differ by π radians.

3. What do the displacement graphs show?

The displacement graphs show sinusoidal motion, with one object reaching its maximum displacement while the other reaches its minimum, reflecting antiphase behavior.

4. How does the velocity graph reflect antiphase motion?

The velocity graph shows sinusoidal curves for both objects, with the peaks and troughs of one object inverted compared to the other.

5. What are the practical applications of this model?

• Understanding wave interference (e.g., sound waves or water waves).
• Studying mechanical systems with antiphase oscillations.
• Modeling synchronized oscillators in engineering systems.

6. Can this model include damping?

Yes, damping can be incorporated by modifying the position functions to include an exponential decay term:

x(t)=Ae−btsin⁡(ωt+ϕ)

7. How can this model be extended?

This model can be extended to analyze interference patterns in two-dimensional systems (e.g., ripple tank simulations) or to study energy transfer in coupled oscillators.