1. What Forces and Torques Act on the Hula Hoop?

To summarize (a more in-depth description is in Written Analysis), the hula hoop while in the air is acted on by a gravitational force, and while contacting the ground but slipping a frictional force accelerates the hula hoop to the left (decreasing the horizontal velocity magnitude) while a torque from that frictional force accelerates the hula hoop clockwise (decreasing the angular velocity magnitude). At some point, the hula hoop stop slipping and now the only forces acting on the hula hoop are the normal and gravitational forces which are now equal to each other, and the hula hoop spins and translates at constant velocities.

2. How do the different energies vary when the Hula Hoop is thrown?

As shown from the graphed data from the video analysis and running the code with different energy graphs, the total energy decreases during the event. This is because an outside force (friction which also creates a torque around the center of mass of the hula hoop) opposes the hula hoop's movement, meaning work is done by friction. Breaking it up by energies, for gravitational potential energy, the hula hoop bounces up and down creating what looks like an absolute value function of a decreasing sin curve. Eventually, the gravitational potential energy goes to 0. For translational kinetic energy, the hula hoop starts by moving right, at one point actually has 0 translational velocities, then moves left. This creates a sort of parabola curve with the minimum of the curve being 0 J that transitions to a flat line when the translational velocity becomes constant. For rotational kinetic energy, the graph gradually declines until the hula hoop stops slipping where the rotational kinetic energy remains constant. (If the hula hoop is thrown with enough horizontal velocity and a low angular velocity, the reverse happens between the two energies, with translational kinetic energy gradually declining to a flat line, and rotational kinetic energy looking like a parabola then a flat line.)

3. What is the minimum angular velocity the hula hoop must be thrown at to return to the person given some magnitude?

Assuming the hula hoop is thrown with the same velocity vector that the hula hoop was thrown in my video analysis, by testing the code, which pretty accurately simulates the motion of the hula hoop, if the initial rotational speed is set at 8.2 rad/s, the hula hoop barely moves back to the start. At 8.3 rad/s the hula hoop rolls forward according to the code. In reality, if the hula hoop were thrown with this velocity unless it was somehow perfectly balanced, the hula hoop would probably fall over. It is hard to work out an actual equation that would determine what this initial rotational speed would be variably based on the bounciness of the hula hoop and the initial velocity vector of the hula hoop because you can not realistically calculate the amount of time the hula hoop is in contact with the ground before it stops slipping. However, it is possible to simulate this in the code, because the code uses the Euler Cromer method which is a simplified version of integration. Integration lets you take the sum of things including the amount of time the hula hoop is in contact with the ground given each step of the hula hoop's path (which is simulated in the code). However, applying some kind of conditional integration is not something I have learned to do, so for now, this is the best I can do.

I am quite proud of my code, making all of the widgets work and making it possible to switch between over 10 different graphs of the code. The one thing that does not work fully is that simulating the same throw in my video analysis in my code has it stop slipping at around 1.7 seconds when according to the real data this happens around 2.4 seconds. This was not something I managed to solve. However, the general movement of the hula hoop looks the same, and the code is able to simulate other throws based on inputs from the user. The code uses physics equations I derived from my video analysis and logic conditions that decide when the hula hoop has a certain torque or force affecting its movement as described in the written analysis. This is what gives it the same general movement as my actual video.