*SADIE
AND LAURA'S CRAZY FACTORS OF FLIPPING WEBSITE*

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**Gymnastics was created by the Greeks for three reasons: the
maintenance of good physical condition, for military training and, it was the
conditioning regimen for athletes (Loken and Willouhby). Sadie performs
gymnastics for one reason only, to analyze for physics purposes. Julie Powell
says, “the physics of gymnastics are dependent on the maneuvers being
performed and the person who is performing them,” thus clearly pointing out
that the skill that Sadie performs cannot be exactly duplicated during
multiple trials nor exactly duplicated by anyone else.

The back tuck which Sadie performs in the video is a rotation done head over heels on an axis from one side of her waist to the other; with the top of her body moving towards the opposite direction in which she was facing (USA Gymnastics). While analyzing Sadie’s back tuck we will find the angular velocity through the change in degrees and time further proving our hypothesis: the tighter the tuck the faster the velocity.

Here is an example of a perfect back tuck:

To find the angular velocity we will use our knowledge of angular rotation. The formula for calculating this velocity is:

Angular velocity: w = q/t

To solve our hypothesis we recorded Sadie’s flip and analyzed it frame by frame on the television. When the flip was recorded we timed the length of time between takeoff and landing and we also recorded the height and distance of the flip, which we found irrelevant in our calculations. From Gerry Smith and the lab he developed we used his idea of placing hip markers to analyze the center of mass position (Smith). In addition to the hip markers we also added markers on Sadie’s knees, shoulders, and ankles. We compared the position of the markers on Sadie to the vertical axis in order to find the angle of rotation. We analyzed frames from when Sadie left the ground to the frame when she landed.

Our physics adventure began when we (Sadie and Laura) traveled to Westside Gymnastic Academy. At the gym we video taped Sadie performing a standing back tuck. Before the taping we had attached markers to Sadie’s hip, shoulder and knee. We placed these markers in order to compare their location to the time and vertical axis. With these variables we were then able to start collecting data. The first data we collected was the change in the angle of the shoulders in comparison with the vertical axis between each frame. We discovered that each frame was 1/30 of a second when viewing the video with a VCR. To find this change in angle we traced the location of Sadie’s shoulder, hip and knees from the television onto paper and used a protractor to measure the angle between the shoulder and the vertical axis.

Here is an example of a tracing we made and the angles we found:

Here is an example of a back flip with the angles drawn in:

To find the change we found the difference of the angles between each frame.

An example would be: In frame
one the angle between the vertical axis and Sadie’s shoulder is 45°. In
frame two the angle is 50°, this creates a difference of 5°. We then changed
5° to radians by multiplying the degrees by π/180 since the units for
angular acceleration are radians per second. With this data we calculated the
angular velocity (ω=θ/t) by using the change in angle and dividing
it by 1/30 of a second, since that is the time of each frame. An example of
this would be taking the angle difference between frame one and two, which was
.0872 radians, and dividing it by 1/30 seconds. The result of this situation
was 2.642 radians/second. The next data we took was the angular velocity of
Sadie’s legs. We used the same procedure to find this velocity as we did for
Sadie’s shoulders except we used the angle difference found between the legs
and the vertical axis in each frame. The last set of data was the change in
angle between Sadie’s shoulders and knees, is the degree of tuck which we
found by calculating the difference of the angles between each frame. We then
made graphs of the data in order to prove our hypothesis.

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Our results have proven our hypothesis, as the angle of tuck decreases, the angular velocity increases. We were able to prove this through analysis of the angle between Sadie’s shoulders and the vertical axis and the angle between her legs and the vertical axis. We found that the angular velocity is always greater when her shoulders or legs were closest to the rest of her body. We also found that the angular velocity was not as high for the shoulders which we believe is because Sadie’s shoulders do not have to cover as much distance as her legs. An example of this is seen between frame one and two where the velocity due to the shoulders is 2.644 radians/second and the velocity of the legs is 7.97 radians/second which is evident in the graphs of these velocities which are to follow. We also made a graph of the tuck angle versus time between the shoulder and knee (θ2). This graph is also included. The slope of this graph is the angular velocity of the tuck. By looking at the change in the tuck angle between frames nine and ten where the angle difference is 18, between frames ten and eleven where the difference is 6, and between eleven and twelve where the difference is 3, we have effectively proven that the tuck does get tighter as the flip goes on. Our graph does not give a perfect example of what a tuck angle would look like; if Sadie were to have performed a perfect back tuck the angle would get bigger before the landing, thus making this graph a more V-shape. Another variable that impacted our data was the location of where we placed the shoulder markers. We placed the markers enough on the arms, that we recorded a variety of movements the arm makes instead of the stationary location of the shoulders. This location of the markers affects our uncertainty along with the questionable accuracy of the VCR’s frame time. We might have been able to better analyze the video if we were able to use a digital program on the computer, but we didn’t have the time or money to buy a cord. Following this page is a copy of our data table and our graphs.

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In the end our data did support our hypothesis that the tighter the tuck
the faster the angular velocity. We ran across many difficulties in our road to
success, many of which we could not fix, so we had to change our focus and
procedure of the project. Our main change in plan was when we decided to
eliminate the calculation of the moment of inertia. We were going to use the
equation L= *I*ω to calculate the
moment of inertia in order to prove the conservation of momentum. The
conservation of momentum states that the bigger the velocity the smaller the
moment of inertia, and the smaller the velocity the bigger the moment of
inertia. We decided to eliminate this from our project because we felt we did
not have an accurate enough way of analyzing the data to thoroughly prove our
point. So our research became only the analysis of the velocity and the degree
of tuck. Our other major change in plan was our usage of digital video software
on the computer to analyze our data. We were not able to use digital software
because we did not have the time or money to order a cord that connected the
digital video camera to the computer. We were counting on this software to turn
our video, since the flip Sadie performed was video taped sideways. Since we had
no digital software to use we had to come up with a creative, yet accurate way
of proving our hypothesis. Our first attempt was unsuccessful. We started the
stop watch at the moment Sadie’s feet left the ground and stopped the video in
some random spot during the flip while also stopping the stop watch. We used
this technique to mark a spot at the beginning, the middle, and the end of the
flip. We realized how inaccurate this was and decided to try using Sadie’s VCR
which had a frame by frame option. Using the frame by frame option we marked
every frame starting from the frame where Sadie left the ground to the frame
where she landed. We also decided to only look at one of the flips we recorded
since one flip took about 20 frames. Some
other difficulties came from Sadie’s back tuck itself. Though Sadie is highly
trained and a ten year veteran of gymnastics her back tuck is not perfectly
symmetrical such as the one in the diagram. The first variable of error came
from the position Sadie landed in. In a perfect back tuck you start and end in
the same position, so Sadie should have landed standing somewhat straight up,
this would have altered our graph of the tuck angle by making the trend line a
V-shape. The other variable that affected our data was the placement of the
markers on Sadie’s shoulders. The markers were placed enough on the arm that
the movement of the arm created some interesting variety of data for the angle
between the shoulders and the vertical axis. If the markers were placed farther
back on the shoulders, to where it didn’t have a variety of movement like the
arm, the data would have been more consistent and predictable. We also could
have improved our whole research by analyzing more than one flip; even though
each flip is different, real scientists usually study experiments by repeating
it over and over.

Though we had some factors that altered our data, we also made a finding that we really didn’t mean to. When we made our graph of the angular velocity of the shoulder angle and the leg angle we noticed that the velocities were very different. The angular velocity of the legs was much higher than that of the shoulders as seen between frame one and two when the velocity of the shoulders was 2.644 radians/ second and the velocity of the legs was 7.97 radians/ second. Through severe scientific thought we figured that this was due to the fact that Sadie’s legs had a longer distance to travel because they are farther from the central axis point on her body. This means that Sadie’s legs had to travel a greater distance than her shoulders in the same amount of time, so the legs had to be moving faster. From this discovery, we thought that if we had the opportunity to really research deeper into the factors of flipping we would have calculated how much more distance the legs cover than the shoulders. Another factor we would have liked to figure out is how much force Sadie lands with and at what point is the most force being exerted on her. This additional information would have opened a whole new episode to the factors of flipping but the information we have gathered about the angular velocity of a standing back tuck is enough to give you a satisfying feeling because we have provided scientific data proving the always assumed theory that tighter the tuck the faster the velocity.

**http://www.usa-gymnastics.com,
a guide to gymnastic news and articles about the sport.**

**www.thinkquest.org,
a website containing the history of gymnastics and information about how the
sport works.**

http://home.nc.rr.com/enloephysics/sports.htm,
**offers us a glimpse into a fascinating world of motion in gymnastics.**

Kinematics
of a Gymnast, **a study done by an Oregon State Professor on the kinematics
of a giant performed on the bars.**

A
Physics Interpretation of The Front Handspring Vault, **a website that
focuses on the dynamics of a front handspring performed on the vault.**

1. Giancoli.
__Physics, Fifth Edition__. 1998

2. Loken, Newton C., and Willoughby, Robert J. __The
Complete Book of Gymnastics__. 3rd ed. NewJersey: Prentice - Hall, Inc., 1977.

3. Powell, Julie. “A
Physics Interpretation of The Front Handspring Vault.” 2001, http://angelfire.com/sc2/physics212.htm

4. Smith, Gerry, Ph.D. “Kinematics of a Gymnast.”
www.orst.edu/instruct/exss323/angular_kinematics_lab/indexa.htm

5. ThinkQuset Inc.
“Gymnastics.” 1995, www.thinkquest.org

6. USA Gymnastics Online. “Technique.” 2002, www.usa-gymnastics.org

7. Woolard, Liz. “Physics of Gymnastics and Cheerleading.”
2002, http://home.nc.rr.com/enloephysics/sports.htm