Force and Acceleration

Objective:

In this exercise you will analyze a video clip.  The clip shows a cart on an air track being pulled by the gravitational force acting on a hanging mass.  You will determine the  position of the cart 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 cart 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 a velocity versus time graph yields the acceleration of the cart.

The board in the clip indicates is that the mass of the cart is mcart = 0.196 kg
The hanging mass has mhanging = 0.007 kg.  
The magnitude of the force acting on the system is the magnitude of the weight of the hanging mass, F = mhangingg
We expect the magnitude of the acceleration to be equal to a = F/m, where m is the total mass that is accelerating, i.e. m = mcart + mhanging.

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.  Choose the pull_1.avi video clip.

Begin

bullet"Play" the video clip.  When finished, "Step up" to frame 0.  In some browsers you have to click "Pause" first..
bulletIn the setup window choose to track the x-coordinate of an object.
bulletClick "Calibrate".  Then click "Calibrate X".
The video clip contains calibration markers.  Each white card is 10 cm = 0.1 m wide and the space between card is 10 cm = 0.1 m wide.  Position the cursor over some marked position in the right part of the frame, for example the left edge of the sixth card from the left, and click the left mouse button.  Then position the cursor over some marked position in the left part of the frame, for example the left edge of the leftmost card, and click the left mouse button again.  This will record the x-coordinates of the chosen positions.  Enter the distance between those positions into the text box in units of meter.  For the example positions, you would enter 1 into the text box.  Click "Done".
bulletClick the button "Click when done calibrating".  A spreadsheet will open up.
bulletClick "Start taking data". 
Pick the point on the cart whose position you will track, for example the little post sticking out of the top of the cart.  Position the cursor over that point.  When you click the left mouse button, the time and the x-coordinate of your chosen point will be entered into the spreadsheet.  You will automatically step to the next frame of the video clip.
bulletRepeat for each frame in the video clip.  Then click "Stop Taking Data".
bulletHighlight and copy your table.  Open Microsoft Excel, and paste the table into an Excel spreadsheet.
bulletYour spreadsheet will have two columns, column A, time (s), and column B, x (m).  Produce a graph of position (vertical axis) versus time. (horizontal axis).
bulletRight-click the data in your position versus time graph and choose "Add Trendline".   Choose Polynomial, Order 2. and click "Display equation on chart".  An equation of the form y = b1x2 + b2 x + b3 will be displayed where b1, b2, and b3 are numbers.  What do the coefficients b1 and b2 tell you?   (For motion with constant acceleration we expect that x changes as a function of time as x = x0 + v0t + (1/2)at2, where a is the acceleration.)
What is the magnitude of the acceleration a of the cart that you obtain from the trendline?
 
bullet

Type =(B3-B2)/(A3-A2) into cell C2.  This yields the average speed of the cart 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 C.  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.

bulletIn cell C1 type v (m/s).
bulletClick data, data analysis, regression.  For the input y range choose column C.  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.
bulletThe 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.  We expect it to be equal to a = F/m.
What is the magnitude of the acceleration a of the cart that you obtain from the regression function (slope of the v vs t graph)?
bulletDo your two values for the magnitude of the acceleration agree within the standard error from the fit?
 
bulletAssuming the masses written onto the board are correct, what value do you expect for the magnitude of the acceleration a?
bulletWhat is the percentage difference between the expected and the measured values??

To earn extra credit add your name and e-mail address to your spreadsheet.  In full sentences answer the tquestion posed.

Save your Excel document (your name_exm4.xlsx).  Go to Blackboard, Assignments, Extra Credit 4, and attach your document.