3. Measure the Initial Length of the Spring:

1. • Before adding any weight, measure the spring's natural length (unstretched length) using a ruler. Record this length as L0​.

4. Add Weights and Measure the Stretch:

1. • Gradually add one weight at a time to the spring.

   • After adding each weight, allow the spring to rest, and measure the new length of the spring L. Record the elongation, which is ΔL=L−L0​, for each corresponding weight.

5. Record the Force Applied:

1. • The force applied to the spring due to the weight is given by F=mg, where mmm is the mass of the weight and g is the acceleration due to gravity (approximately 9.8 m/s²).

   • Ensure you convert the mass units to kilograms (if necessary) and calculate the force in newtons (N). Alternatively, the force can be recorded in kilogram-force (kgw) or gram-force (gw), depending on your preferred units.

6. Repeat for Multiple Weights:

1. • Repeat the process by adding different masses to the spring and recording the corresponding elongation and force.

7. Data Collection Using a Web Camera (Optional):

1. • Set up a web camera to record the spring’s motion and elongation. You can use video analysis software (like ImageJ) to help track the spring's displacement more precisely if needed.

8. Calculate the Spring Constant k:

1. • For each weight, plot the force FFF against the elongation ΔL. The spring constant k is the slope of the linear region of this plot, according to Hooke’s Law F=kΔL.

Procedure:

1. Start the Experiment:

   • Apply a force to the system to initiate the vertical oscillation of the mass attached to the spring. Ensure that the mass is moving vertically up and down in a regular, stable manner.

2. Record Videos:

   • Every 30 seconds, capture a 5-second video of the mass oscillating vertically. This will allow you to analyze the motion at different time intervals.

   • Record the exact time when each video was captured to keep track of the intervals and synchronize the data. In total, record 9 video clips of the mass’s vertical vibration.

3. Transfer to ImageJ for Analysis:

   • After recording the videos, input the video files into the ImageJ software.

   • Use ImageJ to perform frame-by-frame analysis of the motion, extracting the position of the mass at each point in time. This step will allow you to accurately track the mass’s movement and quantify its displacement over time.

4. Analyze Motion Data:

   • From the extracted frames, record the relationship between time and the position of the mass in each video. You can plot the position vs. time graph for the oscillation, as shown in the example image below (not shown here in the text).

5. Further Analysis:

   • With the position vs. time data, analyze the oscillatory motion to find key parameters such as amplitude, frequency, and period. This data can then be used to better understand the dynamics of the vertical vibration.

To analyze position vs. time data and extract the period of oscillation using the Fast Fourier Transform (FFT) method, follow these steps:

1. Introduction to FFT

The Fast Fourier Transform is a numerical algorithm that computes the Discrete Fourier Transform (DFT) of a dataset and provides information about the frequency components present in a signal. When applied to position vs. time data from an oscillating mass-spring system, the FFT will reveal the dominant frequency, from which we can determine the period of oscillation.

2. Method Overview

1. Capture position vs. time data: Use ImageJ or similar software to track the motion of the mass and extract the displacement data as a function of time, x(t).

2. Apply FFT: Decompose the position vs. time data into its frequency components using FFT.

3. Identify the dominant frequency: The highest peak in the frequency spectrum corresponds to the oscillation frequency f.

4. Determine the period: Once the frequency f is found, the period T of the oscillation can be computed using T=1/f.

3. Step-by-Step Process

3.1. Collect Position vs. Time Data

• Record the motion of the mass using a web camera and analyze the video in ImageJ to extract the position x(t) vs. time t data.

• Export the data as a two-column dataset: Time t and Position x(t).

3.2. Preprocessing the Data

 Before applying the FFT:

• Ensure that the data is evenly spaced in time. If the time intervals are not uniform, consider interpolating the data to create evenly-spaced time points.

• Subtract the mean from the position data to remove any DC offset. This can be done by subtracting the average position from each data point

• his step ensures that only the oscillatory components of the motion are emphasized in the FFT.

3.3. Perform the FFT

Now apply the Fast Fourier Transform to the position data xcentered. The FFT algorithm will decompose the data into its frequency components.

You can perform this step using software like Python, MATLAB, or any tool that supports FFT analysis. Below is an example using Python:

import numpy as np

import matplotlib.pyplot as plt

from scipy.fft import fft, fftfreq

# Sample position and time data (replace with your actual data)

t = np.linspace(0, 10, 500)  # time points

x = A * np.cos(2 * np.pi * f * t + phi)  # synthetic position data for demonstration

# Subtract mean to remove DC offset

x_centered = x - np.mean(x)

# Perform FFT

N = len(t)  # number of data points

dt = t[1] - t[0]  # time step

X_f = fft(x_centered)  # FFT of the centered position data

frequencies = fftfreq(N, dt)  # Frequency components

# Take the positive half of the spectrum (since FFT is symmetric)

positive_freq_indices = np.where(frequencies > 0)

frequencies = frequencies[positive_freq_indices]

amplitude_spectrum = np.abs(X_f[positive_freq_indices])

# Plot the amplitude spectrum

plt.plot(frequencies, amplitude_spectrum)

plt.title('Frequency Spectrum')

plt.xlabel('Frequency (Hz)')

plt.ylabel('Amplitude')

plt.grid(True)

plt.show()

3.4. Extract the Dominant Frequency

• From the frequency spectrum plot, identify the peak corresponding to the dominant frequency fpeak​.

• This peak represents the oscillation frequency, as the highest amplitude corresponds to the periodic signal of the mass-spring system.

3.5. Calculate the Period

The period T of the oscillation is the reciprocal of the frequency:

Thus, the period is easily obtained from the dominant frequency component.

Step-by-Step Procedure:

1. Access the PhET Simulation:

   • Open the PhET virtual lab and navigate to the simulation for springs and masses.

   • Select the "LAB" mode to access the full range of simulation tools and settings.

2. Initial Setup:

   • Start by fixing a weight of 50 grams (g) on the mass holder.

   • On the right side, select five different spring constants by adjusting the spring strength from small to larger values. To do this, start with the smallest spring constant (e.g., “small”) and incrementally adjust it to the right for each trial until you have tested five different spring constants.

3. Activate Key Tools:

   • Check the boxes for "Mass Equilibrium" and "Movable Line" on the simulation interface. This will help with alignment and equilibrium position tracking during the experiment.

4. Configure Environmental Settings:

   • Ensure the gravitational setting is set to 9.8 m/s² (the Earth's gravity).

   • Set the Damping to None to eliminate any frictional or damping forces that could affect the measurements.

5. Position the Movable Line:

   • Adjust the Movable Line (the red line) to be directly below the spring when it is at rest, without any mass applied.

   • Once aligned, click the pause button to stop the simulation temporarily.

6. Place the Mass on the Spring:

   • Place the 50g mass on the spring, and you should see the spring stretch downward due to the added weight.

   • Observe a black dashed line indicating the mass equilibrium point in the simulation. Ensure the center of the mass aligns with this dashed equilibrium line.

7. Measure Spring Extension:

   • Press the play button to resume the simulation. Use the ruler tool (available on the right side of the screen) to measure the extension of the spring caused by the mass. This is the displacement xxx, which is the difference between the spring's original length and its stretched length.

   • Record the spring’s extension for this mass.

8. Calculate the Spring Constant k:

1. • Apply Hooke's Law to calculate the spring constant k:
     F=k⋅x

2. Where:

   • F is the force applied by the weight (equal to mg, where mmm is the mass and g=9.8 m/s²),

   • k is the spring constant,

   • x is the measured extension of the spring.

3. Rearrange Hooke's Law to solve for k:
   k=F/x​

9. Repeat for Different Weights and Spring Constants:

3. • Change the mass to 70g, 90g, 110g, and 130g.

   • For each mass, repeat the above steps (place the mass, measure the spring extension, and calculate k).

   • Perform this for each of the five spring constants you have selected (from smallest to largest spring constant).

10. Record and Analyze Results:

3. • After completing the trials with all five masses for each spring constant, record your calculated spring constants k and compare them across different spring settings.

   • Ensure the units are consistent, typically in Newtons per meter (N/m).

Following these steps, you will have determined the spring constant for five different springs using the PhET virtual lab.

Step-by-Step Procedure:

1. Access the PhET Simulation:

   • Open the PhET virtual lab and navigate to the simulation for springs and masses.

   • Select the "LAB" mode to access the full range of simulation tools and settings.

2. Initial Setup:

   • Fix a mass of 50 grams (g) on the mass holder.

   • Adjust the spring constant by selecting five different values, starting from "small" and moving toward larger spring constants. Adjust the spring strength to the right, testing five distinct spring constants.

3. Activate Key Tools:

   • On the right side of the interface, check the boxes for "Mass Equilibrium" and "Movable Line." These tools will help you align the mass at equilibrium and track its motion.

4. Configure Environmental Settings:

   • Set the gravitational constant to 9.8 m/s² (Earth’s gravity).

   • Adjust the Damping to None to eliminate frictional effects and ensure the system behaves like an ideal mass-spring oscillator.

5. Position the Movable Line:

   • Align the Movable Line (the red line) directly below the spring when it is at rest, without any mass applied. This ensures that you start with the spring in its natural length.

   • Pause the simulation using the pause button to stabilize the setup.

   • Switch the simulation to Slow Mode to observe the oscillations better.

6. Place the Mass and Start the Oscillation:

   • Gently place the 50g mass on the spring without stretching it initially.

   • Use the timer tool from the right side of the screen to measure time. Press the timer's start button before initiating the oscillations.

   • Resume the simulation by pressing the play button and letting the spring oscillate vertically.

7. Measure the Oscillation Period:

   • Allow the spring to oscillate freely and measure the time to complete three full oscillations (up and down).

   • Divide the total time by 3 to calculate the period T of one complete oscillation. Record this value.

8. Calculate the Spring Constant k:

   • Using the measured period TTT, calculate the spring constant k using the following equation derived from the oscillation period formula for a mass-spring system:

T²=4π²m/k

9. Rearranging for k:
   
   

k=4π²m/T²

Where:

9. • T is the period of oscillation,

   • m is the mass (in kilograms),

   • π is approximately 3.14159.

10. Repeat for Different Weights and Spring Constants:

    • Change the mass to 70g, 90g, 110g, and 130g.

    • For each new mass, repeat the above steps: place the mass, measure the oscillation period, and calculate the spring constant k.

    • Perform these steps for the five different spring constants (smallest to largest).

11. Record and Analyze Results:

    • After completing all trials, record the calculated spring constants k for each spring and mass combination.

    • Analyze the values to ensure consistency and compare the results across the different spring constants.

Following these steps, using the oscillation period method, you will successfully determine the spring constant k for five different springs.