Angular Kinematics
The giant swing is a good example of angular rotation. Your analysis will be based on figures of a gymnast which were taken with high speed films during competition. A series of images during one revolution of the bar have been combined into a single picture.
The eleven frames included in the figure were uniformly separated in time: 0.15 seconds between each position. The scaling of the figures can be determined by the meter stick which is included in the diagram.
The analysis will involve determination
of the gymnast's angular position, angular velocity, and linear velocity
as a function of time. The reference for the angular measures is in the
vertical direction. As the gymnast 's rotation was in the clockwise direction,
we will take that to be the positive direction: position 1 being nearly
vertical will have an angle of approximately zero. Angles for postion 2,
3, etc. will increase in magnitude until the final position 11, in which
the gymnast has nearly rotated through 360 degrees.
To determine the angular position for each of the 11 positions, use the
hip marker as an estimation of the center of mass position. Using a method
similar to the 2-d introduction lab on the web, determine the coordinates
of the hip marker for each position, and also the bar position. From these
X and Y coordinates, using trigonometry, you will determine the angle from vertical
for each position. For example, at position 3, the hip coordinates are (325,
145) and the bar coordinates are (245,263). From these raw data, you can calculate the differences in the X and Y directions (X-diff and Y-diff). Make sure each difference is a positive number (what mathematical function returns any number as a positive number?). The angle from vertical can
be calculated using the arctangent function of the X-difference (325-245) divided by the Y-difference (263-145), or arctangent (80/118).
This corresponds to an angle of 34.1 degrees from vertical. As you
switch quadrant, you will need to change your formula. Recognize
that the angle needs to continuously increase. In spreadsheets, the number PI is represented as follows: PI(). Do not forget to put the parentheses!
Next, determine the radius of the rotation for each frame (distance to the
center of mass from the bar). Calculate the distance using pixel coordinates
from the screen, then convert to real life dimensions using the meter length
stick shown in the figure. For example, at position 3 the gymnast's radius
is the distance from the bar to the hip marker. Using the coordinates above
in the distance formula, the radius is 142.6 pixels. This converts to 1.27
meters.
Record these angle and radius data into a spreadsheet
configured as the example table.
Next step in the analysis is to determine average angular and linear velocity
during the giant swing. Formulas for calculation of angular and linear velocity
during a rotation can be found in your class notes! Using the first
central difference method, find the angular velocity for positions
2 to 10. Express these angular velocities in radians per second, then calculate
the linear velocity for positions 2 to 10.
Finally, illustrate how angular and
linear velocity changed throughout the giant swing by graphing the relationships
using spreadsheet data:
Based on your measured position
data and calculations, briefly summarize, in your own words, the methodology
and results of this experiment. Discuss those characteristics as found in
your results. Include responses to the following questions in your discussion.
Limit this writing to three pages double-spaced. Attach printouts
of your graphs and spreadsheet to the written paper.