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Tracker Modeling in Pendulum as SHM : Model is fx = -w*w*x where w = 3.724 with x0 = 4.60E-2 and vx = 0

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- Pendulum.mp4

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Author: lookang and jitning
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Document Brief: Modeling Pendulum as Simple Harmonic Motion (SHM) Using Tracker

This document details the use of Tracker software to model the motion of a pendulum as simple harmonic motion (SHM). The mathematical model fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x is applied, where $w$ is the angular frequency, x0x_0x0​ is the initial displacement, and the initial velocity is set to zero.

Purpose:

To verify the suitability of SHM equations for modeling pendulum motion under small-angle approximation and analyze the motion parameters using Tracker’s modeling tools.

Key Features:

• Applying a dynamic particle model to simulate pendulum motion.
• Using the SHM equation fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x to fit experimental data.
• Analyzing displacement ($x$), velocity ($v$), and force over time.
• Validating angular frequency $w$ using theoretical and experimental values.

Study Guide: Modeling Pendulum Motion as SHM

Learning Objectives:

1. Understand the SHM model fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x and its physical interpretation.
2. Analyze pendulum motion under small-angle approximation using Tracker.
3. Compare theoretical predictions of SHM with experimental data.

Step-by-Step Guide:

1. Setup and Calibration:

   • Import the pendulum motion video into Tracker.
   • Set the coordinate system and scale using the ruler shown in the video.

2. Tracker Motion Tracking:

   • Track the position of the pendulum bob frame by frame.
   • Observe displacement values along the $x$-axis.

3. Applying the Model:

   • Open the Model Builder in Tracker.
   • Define the SHM equation fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x, where:
     • $w=3.724$ rad/s (angular frequency).
     • Initial displacement x0=4.60×10−2x_0 = 4.60 \times 10^{-2}x0​=4.60×10−2 m.
     • Initial velocity vx=0v_x = 0vx​=0.
   • Run the model and compare it with tracked motion data.

4. Graphical Analysis:

   • Plot $x$-displacement vs. time and observe its sinusoidal nature.
   • Analyze velocity (vxv_xvx​) and force (fxf_xfx​) graphs to validate SHM behavior.
   • Use the model's angular frequency to calculate the period ($T$) using T=2πwT = \frac{2\pi}{w}T=w2π​.

5. Applications:

   • Compare experimental and theoretical values of $w$ to evaluate modeling accuracy.
   • Extend the analysis to real-world systems, such as pendulum clocks or oscillatory systems.

Tips for Success:

• Ensure the pendulum follows the small-angle approximation (θ<15∘\theta < 15^\circθ<15∘) for SHM to be valid.
• Verify the calibration for accurate measurements.

FAQ: Pendulum SHM Modeling Using Tracker

1. Why is the model fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x used for SHM?

This equation describes a restoring force proportional to displacement, a key characteristic of SHM.

2. What does $w$ represent in the model?

$w$ is the angular frequency, calculated as w=glw = \sqrt{\frac{g}{l}}w=lg​​ for a pendulum under small-angle approximation, where $g$ is gravitational acceleration and $l$ is the pendulum length.

3. Why is the initial velocity (vxv_xvx​) set to 0?

The pendulum starts from rest in this experiment, making vx=0v_x = 0vx​=0 a valid initial condition.

4. How do you calculate the period $T$?

The period is calculated using T=2πwT = \frac{2\pi}{w}T=w2π​, where $w=3.724$ rad/s in this model.

5. Why is the small-angle approximation important?

The SHM model assumes that $\sin (\theta )\approx \theta$, valid only for small angles (θ<15∘\theta < 15^\circθ<15∘).

6. What are the limitations of this model?

• The model assumes no damping (ideal conditions).
• It is accurate only for small angular displacements.

7. What are the practical applications of this model?

This model applies to pendulum clocks, vibration analysis, and understanding oscillatory motion in various physical systems.