EXSS 323
Kinematics of a Gymnast

This lab was developed by Gerry Smith, Ph.D., formerly of Oregon State University, and is used by permission. Portions in italics are additions/modifications to the original and were made by Gary Christopher, M.S., A.T.C.

·         INTRODUCTION

The giant swing is a good example of angular rotation. Your analysis today will be based on figures of a gymnast which were taken from high speed films during a gymnastics competition. A series of images during one revolution of the bar have been combined into a single picture.

·         METHODS

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 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 positions 2, 3, etc. will increase in magnitude till 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 methods similar to previous labs, determine the coordinates of the hip marker for each position, and also the bar position. From these X and Y coordinates, using trigonometry, determine the angle from vertical for each position. For example at position 3, the hip coordinates are (290, 129) and the bar coordinates are (218, 234). The angle from vertical can be calculated using the arctangent function of (290-218)/(234-129) = 72/105. This corresponds with an angle of 34.4 degrees from vertical.

After determining the angle with the vertical for each frame, you must then determine the angular displacement. In the 1st quadrant (upper right), the displacement is the same as the angle with the vertical. In the 2nd (lower right) and 3rd (lower left) quadrants, the angle with the vertical is formed with the 180° axis. The displacement in the 2nd quadrant is 180° minus the angle with the vertical; in the 3rd quadrant (lower left), the displacement is 180° + the angle. Finally, in the 4th quadrant (upper left), the angle with the vertical is formed with the 360° axis; the displacement is 360° minus the angle. Think about this and make sure you understand it!

Next, determine the radius of the rotation for each frame (distance to the center of mass from the bar). Calculate this distance using pixel coordinates from the screen, then convert to real life dimensions using the meter stick length 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 127.3 pixels. This converts to approximately 1.29 meters.

Record these Angle and Radius data into a small spreadsheet configured like this example table.

Next step in the analysis is to determine the angular and linear velocity at the instants shown during the giant swing. Formulas for calculation of angular and linear velocity during a rotation are:

The easiest way to do these computations requires some additions to the spreadsheet. Using first-central difference formulas, find the angular velocity for positions 2 through 10. Express these angular velocities in radians per second then calculate the linear velocity for positions 2 through 10.

Finally, illustrate how angular and linear velocity and radius changed throughout the giant swing by graphing the relationships using the spreadsheet data:

    1. Angular Velocity versus Angle
    2. Linear Velocity versus Angle
    3. Radius versus Angle

Graph all the above relationships on the same graph, with angle on the horizontal axis.

·         ANALYSIS:

Summarize your work in this lab with a brief description of methods used for calculating the gymnast's rotational characteristics. Then discuss those characteristics as found in your results. Include responses to the following questions in your discussion. Attach your graphs and spreadsheet to the printout and return these to your lab instructor at the beginning of the next lab meeting.

    1. Where did the greatest angular and linear velocities occur during the giant swing? If they occurred in different frames, explain why.