Show Me - Atwood

The applet Atwood simulates the motion of two masses connected by a massless, ideal string which passes over a massless pulley.

Preamble

This applet can be used to illustrate the physics of the Atwood Pulley. Among the concepts that can be studied are the tensions in supporting strings as well as energy and energy conservation concepts.

This page is designed to get you started using the applet. The applet should be open. The step-by-step instructions on this page are to be done in the applet. You may need to toggle back and forth between instructions and applet if your screen space is limited.

Contents


Finding the acceleration of the falling masses

  1. The applet does not give the accelerations directly but it does provide sufficient information for you to calculate the acceleration of the masses. The information needed to do this is the time t and the distance traveled by mass 1. This data is given in the upper left corner of the drawing panel. As an example, adjust the size of the masses to be 300 g and 700 g respectively (). Next press the play button () and wait. The masses begin to move. Wait until the animation stops. The displacement of mass 1 and the time taken for this appear in the upper left corner of the panel (see bottom figure).
  2. In this case the time of motion was 0.760 s and the distance that mass 1 traveled was 1.133 m.
  3. If we recall that the distance - time relationship for an object accelerating from rest is and that we can rearrange this to give us , then it is a straight forward calculation to show that acceleration a = 3.92 m/s2.

Tensions and FBD's

  1. A Free Body Diagram (FBD) for each mass can be produced by pressing the Free Body Diagram button (). When you do this the images of the masses will fade slightly and force vectors representing the weight and tension will appear (see figure on the right). The hand indicates that mass 2 is held before release. In this case the tension in the strings is due entirely to the weight of mass 1.
  1. You should note that when you press the play button (), the supporting hand disappears and the masses can now move. The tensions shown in the strings will change - try this with several different mass combinations.
  1. The tensions in the supporting strings are not shown. They can, however, be easily calculated by first finding the acceleration for each mass and then applying Newton's 2nd Law. For example, in the previous discussion of acceleration we determined that mass 1 was accelerating upward at 3.92 m/s2. If we use the FBD for mass 1 (shown on the left) and if we assign up as the positive direction, the following force-equation is implied:
    , and since ,we find that
    Since mass 1 = 0.300 kg and a = 3.92 m/s2, we find that T1 = 4.12 N. We leave it for you to show that T2 is also 4.12 N.

 

Working With Potential Energy

  1. When you run the applet Atwood you will see a horizontal line labeled "Ep Reference". This line is used to define a position at which the two masses have zero potential energy. You can capture this line by positioning the mouse over it and then, holding down the left-mouse button, drag up or down. When you "capture" the line it will fade slightly as shown.
    To illustrate this, adjust the masses so that mass 1 = 300 g, mass 2 = 700 g.

 

EP Reference not yet "captured" by the mouse

EP Reference has been"captured" by the mouse
  1. Each mass has a yellow dot which indicates the center of mass for each body. To see how to use the EP reference line effectively, position the EP Reference line so that it passes through the yellow dot (center of mass) for mass 1.
  1. Press the play button (), and wait until the motion stops and then press the View Graph button (). Produce a graph with time on the x-axis and the potential energy of mass 1 (m1 EP) on the y-axis. You should see a graph very similar to the one appearing on the right. Note that the potential energy for mass 1 starts at zero - just as we would expect since we put the EP Reference line at this point. Also, note that when the motion stopped, mass 1 had ascended to a point 1.133 m above the reference line. Since (where Ep1 is the potential energy of mass 1 and h is the height through which it moved), we can insert the numbers to find that:

Ep1 = (0.300 kg)(9.81 m/s2)(1.133 m) = 3.33 J.

  1. You can verify this calculation by inspecting the graph on the right.

Find the Potential, Kinetic and Total Energy of the System

  1. Set up the Atwood pulley so that mass 1 = 250 g and mass 2 = 750 g (this occurs automatically since the total mass is fixed at 1.00 kg). Position the EP Reference line at the center of mass of the 250 g mass. You should have a set up similar to the one shown on the right.
  2. Press play () and note the distance through which each mass moves when the motion stops. You will see that mass 1 rises 1.134 m while mass 2 drops 1.134 m.
  1. Next, press the View Graph button and prepare 4 graphs(You should see something similar to what appears on the right.):
    • time - mass1 EP
    • time - mass 2 EP
    • time - mass1 Ek
    • time - mass 2 Ek
  2. You may need to resize the graph but when you do you can easily inspect the graph and verify that the values appearing are correct. For example, since mass 1 is 0.250 kg and rose through 1.134 m, it is easy to see that:, = (0.250kg)(9.81 m/s2)(1.134 m) = 2.78 J. By clicking on graph 1 and positioning the mouse over top this graph it is easy to see that after 0.684 s, the potential energy of mass 1 is 2.78 J. You can find the other energies by repeating this process for the appropriate graph.

Defining New Variables to Plot

  1. A powerful feature of the grapher is the ability to create new variables that are not listed in the original drop-down menu of variables to plot. Since we plotted the potential and kinetic energy terms for the two masses in the previous example, it is instructive to ask "What would the sum of all of these terms look like?". To do this, close the graph and press the data collection button (). A drop-down menu appears () - choose "Select Data". A dialogue box like the one shown on the right will appear. Since you want to create an expression which does not appear in those listed, press "Add".
  1. After pressing "OK" (), a new dialogue box opens. Fill in the blank spaces the same way as shown on the right. Be very carefully to type the variables exactly as they appear in the list of available variables. You can only build equations out of the preexisting set of variables. When you are finished, press OK. You have now created a variable called "Total Energy" and it is available for plotting.
  1. Now you are ready to plot "Total Energy". To do this, you will need to add one more equation to the graph. Press the small "+" button at the bottom of the graph panel (see figure on the right). A new graph, labeled "undefined graph" appears at the bottom of the previous list of 4 graphs.
  1. Now, proceed as you would with any other graph. Note that this time when you press the "X axis" or "Y axis" buttons a new variable appears in the list - "Total Energy". Select time for the X axis and Total Energy for the Y axis.
  2. Next, press reset () and then press play (). This will "update" the graph and also use the new variable that you just defined. When finished you should see a graph very similar to what appears on the right.