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Center of Mass Using Video Analysis

One-Dimensional Collisions: When analyzing collisions between several objects, physicists define a point that represents the mass center of the system of objects. Imagine that we have two identical carts on a track, and we are analyzing a collision of the two carts. Where would you expect the center of mass of the two-cart system to be? Draw an "x" on the diagram below.

Copy the folder P105COM Video, located in University Physics Students/Team Physics 311/LabPro Folder, to My Documents. Open the file P1051DCOMEqualMass that is in the P105COM Video folder. The movie shows two identical carts on the top track. For this exercise, ignore the other two tracks.

Click on the Add Point icon, then click on the center of mass of cart 1 (red) in each frame of the movie. Notice the graph that appears on the right side of the movie. Sketch data on the axes below.

Rewind the movie, click on the Set Active Point icon, (the horizontal line advances to the middle dot) and click on the center of mass of cart 2 (green) in each frame. Sketch its graph and label.

Rewind the movie, click on Set ActivePoint (horizontal line advances to the lowest dot) and click on a point where you expect the center of mass to be in each frame. Add this data to your sketch, properly labeled.

How does the shape of the center-of-mass graph differ from the graphs for Cart 1 and Cart 2? Are the lines straight, jagged, curved, or what?

Now consider a system consisting of Cart 1 that has twice as much mass as Cart 2. Where would you expect the center of mass of this two-cart system to be? Draw an "x" on the diagram below.

Open the file P1051DUnequalMass that is in the COM Video folder. The movie shows two carts on the top track, with cart 1 more massive. For this exercise, ignore the other two tracks. Click on Add Point, then click on the center of mass of cart 1 (red) in each frame of the movie. Notice the graph that appears on the right side of the movie. Sketch the data on the axes and label.

Rewind, Set Active Point to the middle dot, and click on the center of cart 2 (green) in each frame. Sketch the new data on the graph, properly labeled. Finally, rewind, Set Active Point to the lowest dot, and click on points where you expect the center of mass to be in each frame. Show this data on the graph also, properly labeled.

Do the center-of-mass points in the graph lie along a straight line, jagged line, curved line, or what?

If you marked the centers of mass correctly, you should have gotten a straight line for the center of mass graph. If you got something else, try different ways of determining where the center of mass is until you get a straight-line graph.

If the graph of the center of mass position versus time is a straight line, what can you say about the velocity of the center of mass?

Physicists define the center of mass of two objects (#1 and #2) with masses m₁ and m₂ at locations x₁ and x₂ to be

                                                            (1)

The velocity of the center of mass is the derivitive of x_COM with respect to time. Take the derivative of equation 1 above and write the result here:

                                                                                                                                                         (2)

If we define the momentum of an object to be its mass multiplied by its velocity, then the total momentum of the two-puck system would be

                                                                                                                       (3)

Compare equations 2 and 3. What can you conclude about how behaves in time compared to how the center of mass behaves?

If the total momentum of a system is a constant, then we say that the total momentum is conserved. It can be shown as a consequence of Newton's laws of physics is that the total momentum of a system is conserved whenever the net external force on the system is zero. What are the external forces in this system, and what is the net external force?

Two-Dimensional Collisions: Open the file P1052DCOM. This shows an air table with two pucks of unequal masses on it. The masses of the Small (black) and Big(blue) pucks are shown on the first frame of the movie. Play the movie to see how they collide. Notice that the centers of the pucks have already been marked, and the y- and x-coordinates of the pucks are already shown on the graphs.

As described in the file, sketch a prediction for the graph of center of mass location as a function of time, one for each graph (y and x coordinates). To do this, use the menu Analyze/Draw Prediction. Don't be too fussy about getting this perfect, but have a prediction in which you have confidence.

Next define formulas for the coordinates of the center of mass using Data/Column Options/X_CoM and Data/Column Options/Y_CoM.

How good were your predictions? If you were seriously off, determine what you did wrong.

Print out the LoggerPro screen and include with the materials that you hand in.