Lab Report

Part 1: Equipment Setup and Data Collection

1. List the equipment used in the experiment:

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2. Describe the process of setting up the experiment. What variables are you controlling, and what variables are you measuring?

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3. Record your initial conditions:

   • Height of the liquid in the straw Z0: _____________ cm

   • Submersion depth h for trial 1: _____________ cm

   • Damping coefficient b for trial 1: _____________

4. Record observations for each of the five trials:

Analyze the video recordings using ImageJ:

• What was the initial position of the liquid surface? _____________ cm

• What was the position of the liquid surface after 5 seconds? _____________ cm

Part 2: Data Processing

Convert your videos to AVI format and extract data.

6. • Describe the steps for converting the videos using FFmpeg. What challenges did you encounter, if any?

Use the Python code to analyze the data.

6. • Record the damping coefficient b for each trial

Part 3: Plotting and Analysis

8. Plot your data:

   • Create a plot of b vs. h (submersion depth vs. damping coefficient).

   • Does the relationship between b and h appear to be linear? Explain.

9. Discuss your results:

   • How does the damping coefficient b change with varying submersion depths?

   • What conclusions can you draw about the fluid resistance in this system?

Objective:

 You are tasked with completing an in-depth analysis of the experiment involving gravity-driven liquid oscillations in a straw. Follow the instructions to collect, process, and analyze the data using both experimental techniques and numerical modeling.

 Instructions:

1. Conduct the experiment and record five trials by varying the height h of the submerged straw.

2. Convert the video files into AVI format and analyze them using the ImageJ software to extract the liquid-level data.

3. Use Python to model the oscillations and fit the data to a Newtonian model.

4. Write a brief report (1-2 pages) summarizing your findings, including plots of the experimental data, fitting results, and discussion.

Problem 1: Fluid Oscillation Dynamics

Consider a straw with a diameter of 8 mm and a length of 20 cm submerged in water. The initial height of the liquid inside the straw is 0.5 cm, and the submersion depth hhh is 10 cm. Using Newton’s second law and hydrostatic pressure principles, derive the equation governing the oscillatory motion of the liquid in the straw.

1. Derive the equation of motion using the following parameters:

   • Gravitational acceleration g=9.8 m/s²

   • Water density ρ=1000 kg/m^3

   • Damping coefficient b0=0.05 Ns/m

2. Simulate the oscillation of the fluid using a Python program. What is the expected period of oscillation for this setup?

Problem 2: Analyzing the Damping Effect

In an experiment, you observe the oscillations of water in a straw and obtain the following damping coefficients b for different submersion depths h:

1. Plot the data: Create a plot of b vs. h. Does the relationship appear linear? If so, find the equation that best fits the data.

2. Interpretation: Explain why the damping coefficient b increases with submersion depth h. What physical factors might contribute to this trend?

Problem 3: Hooke’s Law Approximation

For small oscillations, the behavior of the liquid in the straw can be approximated by Hooke’s law, where the restoring force is proportional to the displacement from the equilibrium position. Assume the liquid height oscillates about an equilibrium position at the surface of the water bath.

1. Derive the small displacement approximation for the liquid oscillation, assuming the fluid level z satisfies 

Using the initial parameters, calculate the natural frequency f0​ of oscillation for small displacements.

1. Worksheet (40%): Completeness and accuracy of theory, calculations, and explanations.

2. Assignment (30%): Detailed report, correct use of formulas, analysis, and discussion of results.

3. Problem Set (30%): Correctness of solutions, step-by-step calculations, and proper explanation.

Each lab group should download the Lab Report Template and fill in the relevant information as you experiment. Each group member should answer the Worksheet, Assignment, and Problem individually. Since each lab group will turn in an electronic copy of the lab report, rename the lab report template file. The naming convention is:

[Short Experiment Number]-[Student ID].PDF

Submit the Lab Report in PDf format