GOAL:

• Overall: To determine how the rotational motion of a fidget spinner works

• Specific: To be able to retrieve data which can help me calculate the equations for theta, omega, and alpha.

Set Up:

1. Using the video of the fidget spinner, I tracked the edge of one of the three tips as it rotated around the center (the bearing).

2. Afterwards, I converted the x and y coordinates into an angle using "atan2(y,x)", which is the arctan of y/x but the function returns an angle from -π to π radians depending on the quadrant.

3. Then, I graphed the Position/Velocity/Acceleration and the Theta/Omega/Alpha graphs in terms of time.

4. Finally, for the Theta/Omega/Alpha graphs, I created a manual fit using equations of rotational motion.

Note: Theta is meant to appear like a quadratic, however because atan2 generates values from -π to π, whenever theta crosses from second quadrant to third quadrant, instead of continuing on (ex: π then 2π then 3π), it returns back to –π. (If you stitch each part on top of each other, there will be an upside down parabola.) We can see that the best fit line (assuming the parts are "stitched" together) follows the equations of rotational motion: θ = θ₀ + ω₀t + 0.5αt². The A value of -0.148 is half of α (-0.296 rad/sec²), so it seems the data is very accurate. Additionally, the B value of 10.3 is very close to ω0 of 10.272 rad/sec. And the C value of -2.27 is almost equal to θ₀ of -2.266 radians.

Note: In the Angular Velocity graph above, there are actually four "lines" generated: the correct one (which is at the top), the one below (can be seen in graph), and two more below the x-axis (manually removed from graph). The reason the graph generates these four "lines" is because the omega graph is generated by using the function "derivative('theta')" . When theta crosses from the second quadrant into the third quadrant, the theta value changes from π to -π, so the function sees the change as -2π, when in reality the fidget spinner doesn't change angle by that much. The function also doesn't look at a single point to calculate the derivative, it looks at 3 points around that point. This results in 3 points per revolution being affected by the second to third quadrant error, resulting in the 3 incorrect lines. Luckily, I have over 1000 data points, so there are still many valid data points that can be used here.

The A value (the slope of the graph), represents the angular acceleration of -0.296 rad/sec². The 10.272 rad/sec (at the end of the equation) represents the initial angular velocity.

Note: The reason there is so much spread in the data is because of the issue with the derivative function in the omega graph. However, alpha does seem to be pretty consistent towards the end, which helped me identify -0.296  rad/sec² as the angular acceleration.