Archimedes' Principle

Objective:

In this exercise you will analyze a video clip.  The clip shows a ball rising in a glass of water.  You will determine the  position of the ball as a function of time by stepping through the video clip frame-by-frame and by reading the time and the position coordinates of the ball off each frame.  You will construct a spreadsheet with columns for time and position and use this spreadsheet to find the velocity as a function of time.  The slope of the velocity versus time graph yields the acceleration of the ball.  You will use this acceleration to determine the mass of the ball. 

Procedure:

To play the video clip or to step through it frame-by-frame click the "Begin" button.  The "Video Analysis" web page will open.  You can toggle between the current page and the "Video Analysis" page by pressing Alt-Tab.  Choose the archimedes_1.avi video clip. 

Play the video clip.  When finished, the video clip will rewind automatically and stop at frame 0.
In the setup window choose to track the y-coordinate of one object.
Go to the "Take Data" window.  Click "Start taking data".  A spreadsheet will open up.  Take data as described in a previous exercise.
Calibrate your data as described in that previous exercise, choose an origin and add a calibrated column.
Highlight your table, click "Edit, Copy" on your browser's menu bar, open Microsoft Excel, and paste the table into an Excel spreadsheet by clicking "Edit, Paste" on Excel's menu bar.
Your spreadsheet will have three columns, time, y1, and y1(m).
  A B C
1 time y1 y1 (m)
2 0 124 0.023
3 0.0667 125 0.024
4 0.1333 126 0.026
4 0.2 129 0.03

 
If your spreadsheet looks similar to the one shown above, type =(C3-C2)/(A3-A2) into cell D2.  This yields the average speed of the ball in the small time interval between the first and the second frame of the video clip.
Copy this formula into the other cells of column D.  This will yield the average velocity during successive time intervals.  If the last entry of column A is in row i, let the last entry of column D be in row i-1.

On the menu bar click tools, data analysis, regression.  For the input y range choose column D.  For the input x range choose the corresponding cells of column A.  Under output options check new worksheet, and under residuals choose residual plots and line fit plots.  Click OK.
The regression function finds the best fitting straight line for your data.  Under SUMMARY OUTPUT, X Variable, you will  find the slope of this line, and the standard error in this slope from the fit.  The slope of the velocity versus time graph is the acceleration.
Now find the mass of the ball using Archimedes' principle. 
buoyant force - mg = ma, 
or 
(rwaterVsphere - msphere)g = mspherea.

Question:

What is the measured acceleration of the sphere?
What is the mass of the sphere?
What would the mass of the sphere be if its average density were equal to the density of water?
What percentage of the volume of the sphere will be above water when equilibrium has been reached?

To earn extra credit add your name and e-mail address to your spreadsheet.  In full sentences answer the questions (in blue) posed above.

Save your Excel document (your name_exm1.xls) and attach it to an e-mail message to mbreinig@utk.edu.