# Rotational Kinetic Energy of Circles

I was always curious about the physics involved in even elementary gymnastics skills. I always enjoyed pommel horse and decided to analyse circles.

I uploaded a video of me performing circles on a mushroom into Logger Pro and calculated the period, and radius of the swing. The period and radius are an average of several points.

The data was gathered from this video.

Next, I created a program in python to perform the calculations for me because I enjoy the organization and it allows me to change variables with ease. I will explain what is going on in the program with subtext, but it is also explained in short-hand in the comments. SI units are used for calculations and outputs, but their US equivalents are written in the text for ease of understanding.

This piece of code sets the variables I calculated for my swing. It includes my height, mass, frequency, period, and the distance from my axis of rotation to my center.

Here is what my motion looks like if approximated as a rod.

It is quite difficult to calculate the moment of inertia of a human body because it is an irregular shape. Moment of inertia is calculated with the sum of the the particles with their relative mass and distance to center of mass. Regular shapes have defined functions, so I modeled myself as a cylinder. I then had to use the parallel axis theorem to calculate my moment of inertia with a non-centered axis, and finally I calculated the rotational kinetic energy.

This code outputs my calculated values.

This code outputs the following:

My moment of inertia rotating about my center is 14.4703125 kg*m**2

My angular velocity is 6.49377854541 radians/s

My moment of inertia rotation about my axis of rotation is 18.9474012 kg*m**2

My total amount of kinetic energy in joules is 798.995988937 J

My total amount of kinetic energy in kilocalories is 0.190964624507 kCal

Next, I decided to examine how mass and height affect my rotational kinetic energy. I went back to my python program and changed the height to someone who is 7 feet tall, but kept my mass the same. Notice that I multiplied d by a factor. This is to keep the distance from the center to the axis of rotation in proportion to the height of that person.

The parameters are put through the same code, but the output values also show the factor by which the rotational kinetic energy has changed.

Output:

My moment of inertia rotating about my center is 21.569906421 kg*m**2

My angular velocity is 6.49377854541 radians/s

My moment of inertia rotation about my axis of rotation is 134.662047421 kg*m**2

My total amount of kinetic energy in joules is 5678.58539628 J

My total amount of kinetic energy in kilocalories is 1.35721448286 kCal

My energy changed by a factor of 7.10715131853

Next, I calculated the energy in someone with the same height as me,  5' 9", but weighing 152.63 pounds. Notice that mass is multiplied by a factor. This factor is the ratio between the tall height and original height. This allows us to determine which factor: mass or height, has a greater effect on rotational kinetic energy.

Output:

My moment of inertia rotating about my center is 17.6682515625 kg*m**2

My angular velocity is 6.49377854541 radians/s

My moment of inertia rotation about my axis of rotation is 23.1347768652 kg*m**2

My total amount of kinetic energy in joules is 975.574102492 J

My total amount of kinetic energy in kilocalories is 0.233167806523 kCal

My energy changed by a factor of 1.221

Conclusions:

Height has a much greater effect on rotational kinetic energy than mass.

It takes about 2 kCal (i.e. nutrional calories, Calories) for me to perform ten circles. This is not to say that I am only burning 2 Calories per ten circles, merely that the rotational kinetic energy is that amount. This model does not account for energy used to keep my body straight, push against gravity, or my arms moving up and down.

Physics can get complicated  quickly and it is the job of the student/physicist to make smart assumptions and apply as accurate a model as possible given the resources and knowledge. For example, it is not entirely accurate to model acceleration as constant velocity for every 0.01 seconds, but it is fairly accurate and accomplishes its task.

Physics is the best subject.