編碼器線路顏色對應Arduino角位

紅-5V

黑-GND

白-D2

綠-D3

Measuring the Effect of the Moment of Inertia on the Period and Torsion Constant and Determining the PLA Torsion Constant

Step 1: Measuring the Mass and Radius of the Disk

1. Place the 3D-printed rotating disk on a scale to measure its mass mmm. Ensure the unit is in kilograms (kg) and record the value.

2. Use a ruler to measure the radius r of the disk. The unit should be in meters (m). Record this value.

3. Note: Ensure all units are consistent (mass in kg and radius in meters) when recording.

Step 2: Calculating the Moment of Inertia

1. The moment of inertia III for a disk is given by the formula

where:

1. m is the mass of the disk (in kg),

1. 2. rr is the radius of the disk (in meters).

   3. Compute the moment of inertia III using the measured mass and radius values. Ensure that the units are kg·m².

Step 3: Setting Up the Experiment Equipment

1. Refer to the provided STL files to 3D print the necessary parts and assemble the experimental setup.

2. Upload the appropriate code to the Arduino system that interfaces with the optical encoder.

3. Ensure the system is properly set up and ready for testing.

Step 4: Testing the Optical Encoder Signal

1. Open the Arduino Serial Plotter.

2. Rotate the disk by a fixed angle θ, and observe the signal from the optical encoder.

3. Check if the signal oscillates up and down in the plot (the red line) when the disk is rotated. This confirms that the encoder is functioning correctly.

4. Once the signal behavior is confirmed, close the Serial Plotter.

Step 7: Calculating the Torsion Constant

1. Using the experimentally determined period T and the moment of inertia III, calculate the torsion constant K of the PLA line.

2. The formula for the torsion constant K is based on the relationship between the period and moment of inertia:

Rearranging this equation to solve for K:

1. Perform this calculation to determine the torsion constant K.

Step 8: Repeating the Experiment with Different Weights

1. Repeat the above steps using three different weight configurations:

   • No additional weights (just the disk),

   • Two weights (symmetrically placed on the disk),

   • Four weights (symmetrically placed on the disk).

2. Ensure the weights are placed symmetrically around the disk’s center to maintain balance during rotation.

3. For each weight configuration, repeat the entire process (steps 1 to 7) to collect the data, perform the analysis, and calculate the torsion constant for each setup.

Measuring the Torsion Constant K of the PLA Line by Analyzing the Relationship Between Angular Acceleration and Angle

Step 1: Measuring the Mass and Radius of the Disk

1. Take the 3D-printed rotating disk and place it on a scale to measure its mass mmm. Ensure the unit is in kilograms (kg) and record the value.

2. Use a ruler to measure the radius rrr of the disk in meters (m). Record this value.

3. Note: Be consistent with the units during measurement and recording (mass in kg and radius in meters).

Step 2: Calculating the Moment of Inertia

1. The moment of inertia III for the disk is calculated using the formula: 

where:

1. m is the mass of the disk (in kg),

2. r is the radius of the disk (in meters).

3. Compute the moment of inertia III using the measured mass and radius values. The unit of I should be kg·m².

Step 3: Setting Up the Experimental Equipment

1. Refer to the provided STL files to 3D print the required components and assemble the experimental setup as per the instructions.

2. Upload the code to the Arduino system, ensuring it is properly connected to the optical encoder for data collection.

Step 4: Testing the Optical Encoder

1. Open the Arduino Serial Plotter.

2. Rotate the disk by a fixed angle θ\thetaθ, and check if the optical encoder signal shows an oscillating pattern (red line) on the plot. This confirms that the encoder is capturing the angular displacement correctly.

3. Once confirmed, close the Serial Plotter.

Step 5: Recording Data in the Serial Monitor

1. Open the Serial Monitor in Arduino.

2. Rotate the disk by a fixed angle θ to initiate the oscillatory motion.

3. Copy the angular displacement data from the Serial Monitor and paste it into LibreOffice Calc.

4. Save the data as a CSV file for later analysis.

Step 6: Plotting Angular Acceleration vs. Angle

1. Use Python to analyze the data and calculate the angular acceleration from the angular displacement values.

2. Plot the graph of angular acceleration α\alphaα against the angle θ.

3. Perform a linear fit on the data using Python. The slope of the linear fit represents the value of KI​, where:

   • K is the torsion constant,

   • I is the moment of inertia.

Step 7: Calculating the Torsion Constant K

1. From the linear fit, determine the slope mmm, which equals KI​​.

2. To find the torsion constant K, multiply the slope by the calculated moment of inertia I

 3. Using this relationship, compute the torsion constant K of the PLA line.

Step 8: Repeating for Accuracy

1. You can repeat the experiment by rotating the disk at different fixed angles θ\thetaθ to confirm the consistency of the results.

2. Ensure that you record data accurately and repeat the analysis to validate the value of the torsion constant.

Step 2: Rotating the Disk and Recording the Oscillations

1. Rotate the disk by a fixed angle θ to start the oscillations.

2. Open the Serial Monitor in Arduino and observe the recorded oscillation data.

3. Copy the oscillation data from the Serial Monitor and paste it into LibreOffice Calc.

4. Save the data as a CSV file for further analysis.

where:

1. • T1/2​ is the half-life (in seconds).

Substitute the measured half-life into the formula and calculate the damping coefficient λ.

Step 5: Repeating the Experiment Without Magnets

1. Remove the magnets and repeat the experiment.

2. Follow the same steps (rotate the disk by a fixed angle, record the oscillation data, and perform a fit using Python) to determine the half-life without magnets.

3. Calculate the damping coefficient λ for the system without the magnets using the same formula.

Step 6: Comparing the Results

1. Compare the damping coefficients λ for the cases with and without magnets.

2. Observe and analyze how magnets' presence affects the oscillations' damping.

   • The system with magnets should exhibit a higher damping coefficient, as the magnetic interaction introduces additional resistance, reducing the oscillation amplitude more quickly.