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Tracker Modeling in 2 Pendulum swinging in out of phase

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- Pendulum_outofPhase.mp4

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Author: lookang model, jitning video
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Document Brief: Modeling Two Pendulums Swinging Out of Phase Using Tracker

This document explores the dynamics of two pendulums swinging out of phase using Tracker software. Out-of-phase motion occurs when two pendulums oscillate with a constant phase difference other than 0° or 180°. The motion is analyzed using position functions in Cartesian coordinates, illustrating their relative displacement and phase shift.

Purpose:

To investigate the periodic motion of two pendulums swinging out of phase, analyzing their synchronization, displacement, and velocity using Tracker’s modeling and graphical tools.

Key Features:

• Position functions applied to describe out-of-phase motion:
  • x1=A1sin⁡(ωt+ϕ1)x_1 = A_1 \sin(\omega t + \phi_1)x1​=A1​sin(ωt+ϕ1​),
  • x2=A2sin⁡(ωt+ϕ2)x_2 = A_2 \sin(\omega t + \phi_2)x2​=A2​sin(ωt+ϕ2​), where ϕ1≠ϕ2\phi_1 \neq \phi_2ϕ1​=ϕ2​.
• Graphical representation of displacement ($x,y$) and phase differences between pendulums.
• Verification of synchronization and phase shift through experimental data.

Study Guide: Modeling Pendulums in Out-of-Phase Motion

Learning Objectives:

1. Understand how to model and analyze pendulums swinging with a phase difference.
2. Analyze displacement vs. time graphs to observe phase shifts.
3. Verify the relationship between position functions and experimental motion tracking.

Step-by-Step Guide:

1. Setup and Calibration:

   • Import the video of the two pendulums swinging out of phase into Tracker.
   • Calibrate the scale using the visible ruler for accurate measurements.

2. Tracking Motion:

   • Track the position of both pendulums frame by frame.
   • Record displacement ($x$) data for each pendulum along the horizontal axis.

3. Define Position Functions:

   • Open Tracker’s Model Builder.
   • Define the position functions:
     • For Pendulum 1: x1=A1sin⁡(ωt+ϕ1)x_1 = A_1 \sin(\omega t + \phi_1)x1​=A1​sin(ωt+ϕ1​),
     • For Pendulum 2: x2=A2sin⁡(ωt+ϕ2)x_2 = A_2 \sin(\omega t + \phi_2)x2​=A2​sin(ωt+ϕ2​),
     • where:
       • A1=0.032A_1 = 0.032A1​=0.032, ϕ1=3.048\phi_1 = 3.048ϕ1​=3.048 rad,
       • A2=0.032A_2 = 0.032A2​=0.032, ϕ2≠ϕ1\phi_2 \neq \phi_1ϕ2​=ϕ1​ rad (phase difference).
     • $y=0$ for both pendulums (horizontal motion only).

4. Graphical Analysis:

   • Plot x1x_1x1​ and x2x_2x2​ vs. $t$:
     • Observe periodic oscillations with a clear phase difference between the pendulums.
   • Verify the phase shift by comparing the peaks and troughs of both graphs.

5. Applications:

   • Study coupled oscillatory systems where elements oscillate with a fixed phase difference.
   • Analyze the effects of damping or external forces on out-of-phase motion.

Tips for Success:

• Ensure precise tracking of both pendulums to capture the phase shift accurately.
• Validate position function parameters by fitting theoretical equations to experimental data.

FAQ: Two Pendulums Swinging Out of Phase

1. What does out-of-phase motion mean?

Out-of-phase motion occurs when two pendulums oscillate with a constant phase difference (ϕ1≠ϕ2\phi_1 \neq \phi_2ϕ1​=ϕ2​), meaning their peaks and troughs occur at different times.

2. How are the position functions defined?

• Pendulum 1: x1=A1sin⁡(ωt+ϕ1)x_1 = A_1 \sin(\omega t + \phi_1)x1​=A1​sin(ωt+ϕ1​),
• Pendulum 2: x2=A2sin⁡(ωt+ϕ2)x_2 = A_2 \sin(\omega t + \phi_2)x2​=A2​sin(ωt+ϕ2​), where A1A_1A1​ and A2A_2A2​ are amplitudes, $\omega$ is angular frequency, and ϕ1,ϕ2\phi_1, \phi_2ϕ1​,ϕ2​ are phase offsets.

3. What does the displacement graph show?

The displacement graph shows periodic motion for both pendulums, with a consistent phase shift between their oscillations.

4. Why do the pendulums oscillate out of phase?

Out-of-phase motion may result from initial conditions where the pendulums are displaced with different phase angles.

5. How can phase differences be measured?

Phase differences (ϕ2−ϕ1\phi_2 - \phi_1ϕ2​−ϕ1​) can be measured by comparing the time intervals between peaks or troughs of the displacement graphs.

6. What are the practical applications of this model?

• Understanding coupled oscillatory systems with phase shifts, such as vibrating strings or coupled pendulums.
• Analyzing phase synchronization in mechanical or electrical oscillators.

7. Can damping or external forces affect the phase difference?

Yes, damping or external forces can alter the amplitudes and phase difference, which can be included in extended models.