That's a fascinating experiment setup! Using Arduino and IR sensors, adding a modern, precise way to measure the pendulum's period. 3D printing parts for adjusting the string length and mass gives you control and repeatability, critical in analyzing the relationship between length, mass, and period.

 Here’s how your experiment should align with the theoretical background:

1. Period vs. Length: The period T of a pendulum is theoretically related to the length l by the formula:

This means that the period should increase with the square root of the length, and it is independent of the mass for small angles. Your setup will allow you to test this dependency experimentally.

2. Mass Dependence: According to theory, for a simple pendulum (under ideal conditions), the mass does not affect the period. This is a direct consequence of the nature of gravitational acceleration being independent of mass. With your setup, you can experimentally verify this by varying the mass and seeing if the period remains constant for different masses.

Using the Arduino and IR sensors to record the exact time period for each pendulum swing and systematically varying the length and mass using 3D printed parts will provide you with data that you can analyze to plot the relationships between these variables. Once your data is gathered, you can compare it against the theoretical models.

In this experiment, you will explore how the length of a pendulum and its mass affect its period of oscillation. Using an Arduino with an IR emitter and receiver for precise timing, along with 3D-printed components to adjust the pendulum's length and mass, you'll collect data to analyze these relationships.

Step 1: Setup of Equipment and Circuit

Begin by assembling all the necessary equipment:

• Pendulum Apparatus: Set up the pendulum stand to ensure stability. Use the provided 3D-printed parts to adjust the length and mass of the pendulum.

• Optical Path: Install the IR emitter and receiver in the correct positions to detect the pendulum as it passes through its lowest point. Refer to the "Setup & STL" guidelines for detailed assembly instructions.

• Electronic Circuit: Assemble the circuit according to the "Electronic Components & Circuit Diagram." Ensure all connections are secure and the Arduino board is properly connected to the IR sensors and OLED display.

Step 2: Preparing the Pendulum

• Measuring Mass: Use a scale to accurately measure the lead ball's mass (pendulum bob). Record this value for your data.

• Assembling the Pendulum: Attach the lead ball to a length of cotton thread. Secure the thread to the pendulum stand using a screw, ensuring that the pendulum can swing freely without obstruction.

Step 3: Calibrating the IR Sensor

Before experimenting, calibrate the IR sensor to ensure accurate period measurements:

• Uploading Calibration Code: Upload the "AnalogRead" code to the Arduino. This code will help you determine the sensor's reading range.

• Recording Sensor Values: Observe the maximum and minimum values of AnalogRead when the pendulum is at rest and when it interrupts the IR beam. Record these values.

• Updating Pendulum Code: Modify the "Pendulum" code by setting the minimum variable to the minimum value you recorded. This step calibrates the sensor to detect the pendulum's passage accurately.

• Uploading Updated Code: Upload the modified "Pendulum" code to the Arduino.

Step 4: Setting the Pendulum Length

• Measuring Length: Measure the length of the pendulum (L) from the pivot point to the center of mass of the lead ball using a meter stick or measuring tape. Record this value.

• Updating Code with Length: In the "Pendulum" code, update the variable L with the measured length (ensure units are consistent, typically in meters).

• Uploading Code: Upload the updated code to the Arduino to apply the new pendulum length.

Step 5: Conducting the Experiment

• Starting the Pendulum: Displace the pendulum bob slightly (keeping the angle less than 15 degrees to satisfy the small-angle approximation) and release it without pushing.

• Recording Oscillations: Allow the pendulum to swing back and forth. The Arduino will detect when the pendulum passes through the IR beam.

• Observing Data: Let the pendulum complete 10 full oscillations. The OLED screen will display the following data:

  • Length (length): The length of the pendulum.

  • Period (Period): The time for one complete oscillation.

  • 4π²L: A calculated value based on the pendulum length.

  • T²: The square of the period.

Step 6: Repeating Measurements

• Multiple Trials: Repeat the measurement process three times for the same pendulum length to ensure accuracy. After each trial, record the displayed values.

• Calculating Averages: Compute the average of the period and other recorded values from the three trials.

• Resetting if Necessary: If you notice any irregularities (e.g., the pendulum doesn't swing smoothly), press the Reset button on the Arduino and restart the measurement.

Step 7: Varying Pendulum Length

• Adjusting Length: Adjust the cotton thread by changing the pendulum's length. Refer to the "Method for Changing Pendulum Length" for guidance.

• Updating Code: Measure the new length, update the L variable in the "Pendulum" code, and upload it to the Arduino.

• Repeating Measurements: Experiment again, following Steps 5 and 6 for each new length. Collect data for multiple lengths to analyze the relationship between length and period.

Step 8: Comparing Experimental and Theoretical Values

• Calculating Experimental g: Use the formula derived from the period of a simple pendulum to calculate the experimental acceleration due to gravity:

Where:

• L is the pendulum length.

• T is the average period from your measurements.

Calculating Percentage Error: Compare your experimental g value with the theoretical value g=9.8 m/s²:

Step 9: Graphical Analysis

• Plotting the Graph: Create a graph with T² on the x-axis and 4π²L on the y-axis.

• Analyzing the Slope: The relationship between T² and L is linear, and the slope of the line should be equal to g:

• Determining ggg from the Graph: Use the slope of the best-fit line to calculate the experimental value of g.

• Comparing with Theoretical ggg: Calculate the error between the graphical g value and the theoretical value as done in Step 8.

Additional Investigation (Optional): Mass Dependence

• Testing Different Masses: Replace the lead ball with masses of different values while keeping the length constant.

• Repeating Measurements: Conduct the experiment for each mass and record the periods.

• Analyzing Mass Effect: Determine if the period changes with mass. According to the theory, the period should be independent of mass for a simple pendulum.

We can definitely simulate the motion of a damped pendulum! In a damped pendulum, the oscillations gradually decrease over time due to friction or air resistance. The equation of motion for a damped pendulum is given by:

Where:

• θ is the angular displacement.

• b is the damping coefficient.

• m is the mass of the pendulum bob.

• g is the acceleration due to gravity.

• l is the length of the pendulum.

For small angles, sin⁡θ≈θ, and the equation becomes:

This equation can be solved numerically using methods like the Euler-Cromer method or Runge-Kutta method.

The driven pendulum motion introduces an external periodic driving force into the system. The equation of motion for a driven damped pendulum is given by:

Where:

• θ is the angular displacement.

• b is the damping coefficient.

• m is the mass of the pendulum bob.

• g is the acceleration due to gravity.

• l is the length of the pendulum.

• Fdrive​ is the amplitude of the external driving force.

• ωdrive​ is the angular frequency of the driving force.

This system exhibits interesting behavior depending on the driving force frequency and amplitude, including resonance and chaotic motion under certain conditions.