About

Tracker Modeling in Light Damping in Pendulum

For Teachers

- Ping Pong (SHM).mp4

Credits

Author: lookang model, jitning video
Contact: This email address is being protected from spambots. You need JavaScript enabled to view it.

Document Brief: Modeling Lightly Damped Pendulum Motion Using Tracker

This document provides an analysis of a pendulum undergoing light damping, modeled using Tracker software. The focus is on the effect of damping on the amplitude of oscillation over time, without significant alteration to the pendulum’s period. The damping force is modeled as proportional to velocity, Fd=−b⋅v, where $b$ is the damping coefficient.

Purpose:

To explore the motion of a lightly damped pendulum, analyze how damping affects amplitude decay, and validate experimental data against the theoretical damping model.

Key Features:

• Motion analysis using Tracker’s displacement tracking and modeling tools.
• Application of the damping force equation Fd=−b⋅v
• Graphical analysis of amplitude decay and its impact on the oscillatory motion.

Study Guide: Modeling Light Damping in a Pendulum

Learning Objectives:

1. Understand the effect of light damping on a pendulum’s motion.
2. Analyze how damping reduces amplitude over time while maintaining the period.
3. Validate the damping coefficient $b$ using experimental data.

Step-by-Step Guide:

1. Setup and Calibration:

   • Import the pendulum motion video into Tracker.
   • Calibrate the scale using visible objects (e.g., a ruler) for accurate measurements.

2. Tracking Motion:

   • Track the pendulum bob’s position frame by frame along the horizontal axis ($x$).
   • Record displacement ($x$) vs. time ($t$) data for several oscillation cycles.

3. Apply the Damping Model:

   • Open Tracker’s Model Builder.
   • Define the position function: x(t)=A⋅e−bt/2m⋅cos⁡(ωt+ϕ)
     • $b$: Damping coefficient,
     • $m$: Pendulum mass,
     • $\omega$: Angular frequency ω=gl−(b2m)2
     • ϕ: Phase offset.

4. Graphical Analysis:

   • Plot $x$-displacement vs. $t$:
     • Observe how the amplitude decreases exponentially while oscillations continue.
     • Verify the period remains nearly constant despite damping.
   • Analyze how the damping coefficient $b$ affects the rate of amplitude decay.

5. Applications:

   • Explore the role of damping in engineering systems (e.g., shock absorbers).
   • Study the effect of light damping on clocks or oscillatory systems.

Tips for Success:

• Ensure accurate tracking of the pendulum’s displacement over multiple cycles.
• Validate theoretical damping parameters with experimental data.

FAQ: Lightly Damped Pendulum Modeling

1. What is light damping?

Light damping occurs when the damping force reduces the amplitude of oscillations gradually without significantly altering the period of motion.

2. How is damping modeled?

Damping is modeled using the equation:

Fd=−b⋅v

where $b$ is the damping coefficient and $v$ is the velocity.

3. How does damping affect the motion?

Damping causes the amplitude of oscillations to decrease exponentially while leaving the period relatively unchanged for light damping.

4. How can the damping coefficient ($b$) be determined?

The damping coefficient can be calculated by fitting the experimental amplitude decay to the theoretical model:

x(t)=A⋅e−bt/2m

5. Why is the period unchanged in light damping?

For small damping, the term b2m\frac{b}{2m}2mb​ is negligible compared to gl\frac{g}{l}lg​, so the angular frequency ($\omega$) and period remain nearly constant.

6. What are the practical applications of this model?

• Understanding damping in clocks or pendulum-based devices.
• Designing dampers in engineering systems like buildings and vehicles.
• Exploring the effect of air resistance on oscillatory motion.

7. How does this differ from heavy damping?

In heavy damping, the damping force is large enough to prevent oscillatory motion, resulting in aperiodic decay towards equilibrium.