This move has always made me think about physics. For those unfamiliar with gymnastics, the kip is essentially a move where a gymnast goes from a position hanging below the bars to one where he or she has the bar at waist level. Here is a super short video example.
The cool thing is that the gymnast starts in a position with low potential energy and ends at a higher potential energy (here I mean gravitational potential energy in the Earth-gymnast system). How does this work? Clearly the gymnast must do some work, but her arms don't even bend. I find this intriguing.
The Physics
Here is what I am going to do. I am going to look at the whole thing in terms of energy. First, some physics. Suppose I model the gymnast as three rigid rectangles that can rotate. One stick will be the legs, one the torso, and one the arms. Here is a diagram.
Each piece does three things. First, the center of mass moves. Second, it can rotate. Third, the object's height can change. Each one of these things can be considered a type of energy. For the motion of the center of mass, there is plain old translational kinetic energy.
Not much needs to be said about the kinetic energy. For the rotation of the object (about the center of mass), there is rotational kinetic energy. It has the form:
Here, I is commonly called the moment of inertia. I like to call it the 'rotational mass'. Essentially, it is a measure of how the mass is distributed around the axis of rotation. For this case, this will be for a rod rotated about its center and it has a value of:
L is the total length of the rod and of course mis the mass. The angular speed is represented by the ω. For the last part of the energy, there is the gravitational potential energy for the object-Earth system (it takes two to make a potential). Near the surface of the Earth, this is just proportional to height. It doesn't really matter where you measure the height from since the only thing that counts is the change in energy. The potential energy can be written as:
So, each piece can have energy. Then I can look at the total energy of the gymnast as the sum of kinetic (both kinds) and potential energies. How do you change this total energy? Work. The work-energy principle says:
Where does the work come from? It comes from the gymnast's muscles. Either that, or some Jedi is near by exerting a force (and thus work) on her.
Assumptions
Here comes the spherical cow part. Essentially, a human is very complicated to model. It is so complicated that I am not going to do that (see above with the three rods). Also, there are some other assumptions I need to make.
- Mass distribution. I could find some way to determine the mass of her arms and legs and the center of mass for each of these. However, I am not going to do that. Instead, I am going to make the following assumptions. Each piece has the center of mass in its center. The arms are 1/6th of the total mass. The torso is half the total mass and the legs are 1/3rd of the total mass.
- I am going to assume that only these three parts move.
The Data
As usual, I used Tracker Video Analysis to get data from the video above. I marked spots for her shoulders, hips, and feet. From the x-y-values of those locations, I could get the x-y center of mass of each part (assuming her hands were at the origin). That part wasn't too difficult.
In order to get the kinetic energy, I needed the velocities. Also, I wanted to look at how the energies changed with the motion of the gymnast. I couldn't figure out a simpler way to make an animated graph along with the movie than to use Logger Pro (which also does video analysis and is almost free - relatively cheap). Also, Logger Pro has a nice smoothing function to make the derivative data look a little nicer.
Here is that data. The three lines are the total energy (orange), total kinetic energy (purple), and total potential energy (red). This video is not in real-time, instead it is me stepping through each frame. Check it out.
First interesting point. In the initial glide move, the total energy actually decreases. Well, I am not surprised that it doesn't increase (the gymnast isn't really doing much but swinging). I guess the decrease in energy is likely due to frictional loses at the bar itself. When she gets to the end of this glide - she looks like this:
From this point, the total energy starts to increase. Here is where the work is done. The first part is when she lifts her legs. This actually seems to do two things. Not only does it increase the center of mass (and thus increase the gravitational potential energy), it also increases her rotational rate (which increases her kinetic energy). Here is a plot of all the energies (along with the total energy).
The important point is that the steepest part of the total energy curve is when she is both moving her legs back down and moving her arms down. This brings up a very key point. I had originally assumed (because I really don't know that much about gymnastics) that this move was mostly in the legs and stomach. Now I think there is a big part in the shoulders. The gymnast really has to have good upper body strength to push her arms down (and increase her center of mass) at the end of this move.
Power
What is the average power this particular gymnast has to produce to make this move? Looking at the graph above, there is a total change in energy of about 176 Joules. This change in energy (and thus the work) takes place in about one second. The power would be:
For a 27 kg girl, that is pretty good. If this were ESPN Sport Science - I would say that is more power per kg than the DEATH STAR!!! But I am not going to do that. No. Let me just say that you have to be pretty strong to do this move. I surely can't do it.
What matters in a kip?
Have I answered my question? Not fully. Let me say arms and shoulders seem to be more important than I thought. Am I finished? Of course not. There is a next step (isn't there always?). What I need to do is to make a simulation of a kip. Take those three rods and model little motors between the arms-torso and one between the torso-legs. Then I can see what would happen if the gymnast doesn't pull up her legs quick enough. I can see what happens if the shoulder 'motor' isn't strong enough. Sure it will be tough to model, but it will be fun.
Acknowledgements: First, a big thanks to Bruce McGartlin of NorthShore Gymanstics for helping me make this video. Also, thanks Abby (you know who you are).
