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Document Brief: Single Swing Pendulum Modeled as SHM Using Tracker

This document explains the modeling and analysis of a single swing pendulum motion using Tracker software. The pendulum is modeled as simple harmonic motion (SHM) with the restoring force fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x. Key parameters include angular frequency ($w$), initial displacement (x0x_0x0​), and initial velocity (vxv_xvx​).

Purpose:

To analyze the pendulum motion under SHM conditions and verify theoretical predictions through experimental tracking and modeling.

Key Features:

• Application of SHM model fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x for pendulum motion.
• Motion analysis using tracked displacement and velocity over time.
• Validation of SHM equations with experimental data.

Study Guide: Modeling Single Swing Pendulum as SHM

Learning Objectives:

1. Understand how SHM equations describe pendulum motion under small-angle approximation.
2. Analyze displacement and velocity graphs for SHM characteristics.
3. Verify the angular frequency ($w$) using experimental data.

Step-by-Step Guide:

1. Setup and Calibration:

   • Import the video of the pendulum’s single swing into Tracker.
   • Calibrate the coordinate system using the ruler in the video for accurate measurements.

2. Tracker Motion Tracking:

   • Track the pendulum bob’s position frame by frame.
   • Observe the displacement ($x$) data along the horizontal axis.

3. Apply SHM Model:

   • Open Tracker’s Model Builder.
   • Define the SHM equation fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x, where:
     • $w=3.136$ rad/s (angular frequency).
     • Initial displacement x0=1.90×10−2x_0 = 1.90 \times 10^{-2}x0​=1.90×10−2 m.
     • Initial velocity vx=−0.14v_x = -0.14vx​=−0.14 m/s.
   • Compare the model's predictions with the tracked data.

4. Graphical Analysis:

   • Plot displacement ($x$) vs. time ($t$):
     • Observe the sinusoidal nature characteristic of SHM.
     • Measure the amplitude and period to verify $w$ using T=2πwT = \frac{2\pi}{w}T=w2π​.
   • Analyze velocity (vxv_xvx​) vs. time for phase relationships between displacement and velocity.

5. Applications:

   • Extend the analysis to study damping or other external forces acting on the pendulum.
   • Use the model to design pendulum-based systems such as clocks.

Tips for Success:

• Ensure the pendulum operates under the small-angle approximation (θ<15∘\theta < 15^\circθ<15∘) for SHM validity.
• Double-check the calibration for consistent data measurements.

FAQ: Pendulum SHM Modeling for Single Swing

1. Why is the pendulum motion modeled using fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x?

The restoring force fx=−w2⋅xf_x = -w^2 \cdot xfx​=−w2⋅x describes SHM, where the force is proportional to displacement and acts towards the equilibrium position.

2. What is the significance of angular frequency ($w$)?

Angular frequency $w=3.136$ rad/s determines the oscillation speed and is related to the period $T$ by T=2πwT = \frac{2\pi}{w}T=w2π​.

3. Why is the initial velocity (vxv_xvx​) nonzero?

The pendulum starts with a slight horizontal velocity (vx=−0.14v_x = -0.14vx​=−0.14 m/s), which is factored into the model to improve accuracy.

4. How do displacement and velocity graphs relate to SHM?

• Displacement vs. time graph is sinusoidal, showing periodic motion.
• Velocity vs. time graph is also sinusoidal but shifted by a phase of $\pi /2$, indicating that velocity is maximum when displacement is zero.

5. What assumptions are made in this analysis?

• Small-angle approximation: $\sin (\theta )\approx \theta$.
• Negligible air resistance or damping forces.

6. Can this model handle damping or non-ideal conditions?

Not directly. This model assumes ideal SHM. Additional terms for damping or external forces can be added to refine the model.

7. What are real-world applications of this analysis?

Understanding pendulum dynamics is essential in designing clocks, seismometers, and studying oscillatory systems in physics and engineering.