Lab Report

Objective:
Analyzing the vertical motion of a mass-spring system and calculating the spring constant k using experimental data and Fast Fourier Transform (FFT) methods.

Part 1: Pre-Lab Preparation

1. Understanding SHM Basics
   a. Define Simple Harmonic Motion (SHM).
   b. Write the equation of motion for a mass-spring system undergoing vertical SHM.
   c. Define the spring constant k, and explain its role in determining the motion.

2. Key Equations
   a. Hooke’s Law:

b. SHM equation:

Where ω is the angular frequency.

3. Tools You Will Use
List the tools and software you will use in this experiment (e.g., weights, ruler, web camera, ImageJ, FFT).

Part 2: Experimental Procedure

1. System Setup
   a. Attach a spring to a fixed support.
   b. Measure the natural length of the spring L0.
   c. Attach a weight of known mass m, and measure the stretched length of the spring. Calculate the displacement x.

2. Start the Oscillation
   a. Displace the mass slightly from its equilibrium position and release it to begin oscillation.
   b. Record the motion using a web camera for a few complete oscillations (at least 3 cycles).

3. Video Analysis Using ImageJ
   a. Import the video into ImageJ software.
   b. Track the position of the mass frame by frame to obtain the position vs. time data.
   c. Export the position vs. time data for further analysis.

4. Data Analysis Using FFT
   a. Use Python or other data analysis software to perform a Fast Fourier Transform (FFT) on the position vs. time data.
   b. From the FFT, identify the dominant frequency f.
   c. Calculate the period T=1/f and use it to determine the spring constant k using:

Part 3: Post-Experiment Questions

1. What is the role of the spring constant k in the oscillatory motion of the mass?

2. How did the FFT method help determine the period and frequency of the oscillation?

3. Compare the spring constant calculated using Hooke’s Law and the value obtained using the FFT method. Were the values consistent? If not, explain possible reasons for the discrepancy.

1. Experimental Setup:
   Could you provide a detailed description of the setup, including the spring, mass, camera positioning, and software used for analysis?

2. Data Collection:
   Attach your position vs. time data extracted from ImageJ. Provide clear, labeled graphs showing the oscillation over time.

3. FFT Analysis:
   Include a plot of the frequency spectrum obtained from FFT. Mark the dominant frequency and explain how you determined the period from this analysis.

4. Spring Constant Calculation:
   Calculate the spring constant k using Hooke’s Law and the FFT method. Compare and analyze the results.

5. Conclusion:
   Summarize your findings, focusing on the accuracy and reliability of using FFT for this experiment. Discuss any challenges you encountered and how you overcame them

1. Problem 1: Hooke's Law Application
   A mass of 0.5 kg is hung from a spring, and the spring stretches by 0.2 meters.
   a. Calculate the spring constant k.
   b. If the same spring is used with a 1 kg mass, what would the new displacement x be?

2. Problem 2: Oscillation Period Calculation
   A mass of 0.2 kg is attached to a spring with a spring constant of 100 N/m.
   a. Calculate the period of oscillation T of the mass-spring system.

3. Problem 3: FFT Frequency Analysis
   During an experiment, you record the oscillations of a mass-spring system and use FFT to identify the dominant frequency of the motion as 0.5 Hz.
   a. What is the period T of the oscillation?
   b. If the object's mass is 0.1 kg, calculate the spring constant k.

4. Problem 4: Energy in SHM
   A spring with a constant of 150 N/m is stretched by 0.05 m from its equilibrium position.
   a. Calculate the potential energy stored in the spring at this displacement.
   b. If the mass attached to the spring is released, what is its maximum kinetic energy during the oscillation?

5. Problem 5: Damping in Real-World Systems
   In real-world applications, damping affects the oscillations of a system. Explain how adding damping to the mass-spring system would change the motion and how you could identify this using the FFT method.

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