N otes on Chapter 6 B
pp 54 - 55 of your lab book

A multiburst image of today's apparatus (exaggerated angle)

• Velocity is the distance traveled per time elapsed.
• Acceleration is the change in velocity per time.

The above photo shows multiple images taken of a cart sliding down an air track. The images were taken 1/30 seconds apart and combined. You can clearly see that there is an acceleration because the distance travelled by the cart in the 1/30 second interval changes with time. If you wanted to find the acceleration of this cart, you would plot the distance travelled by the cart as a function of time. The resultant graph could be fitted to the quadratic equation s = v _0xt + 1/2 a_x t².

But since we don't all have digital cameras, we will use another method to keep track of the position of the cart versus time.  We will use an electronic timer which will give a spark every 0.1 seconds.  Whenever there is a spark, a circular burn will show on a white tape which we will attach to the air track.

To accelerate the cart, you will tilt the track. Theoretically, the acceleration is:

a_x = g sin

(where is the angle to which you are raising the track). Since we will find the acceleration of the cart, a_x, experimentally, we can find the value g, the gravitational acceleration from our data.

In the previous term, students were not able to find the gravitational acceleration g to within tolerance. Rather than complaining about the equipment, let's view this as the excellent learning opportunity it is! Let's discuss the uncertainties in our measurements:

1) Is there a systematic error in the equipment? (Is there anything you can think of that would consistently decrease or increase the acceleration of the cart?)



2) What is the limiting factor? (This is the value with the largest fractional uncertainty).  Can you make this smaller to decrease the uncertainty?



3) The timer used to generate the sparks may be inaccurate. The timing mechanism is an analog circuit rather than a more reliable quartz-based digital circuit. Can you think of a way to determine whether or not this timing mechanism is accurate or not?

Procedure:

Safety notice: The spark timer operates at high voltage. Do not touch any metal part of the apparatus when the red button is pressed.

1. Turn on the air source attached to the air track.
2. Balance the air track by adjusting the wing nuts under the stand.  To balance the track:
   
1. Set the cart moving at a very slow speed.
   
2. If the cart slows down and turns in the other direction, increase the height of the side which the cart is now going.
   
3. Repeat steps 1 and 2 until the cart stops turning around.
   
4. Set the cart moving slowly in the opposite direction.
   
5. If the cart turns around, decrease the height.
   
6. Repeat steps 1-5 until the cart does not turn around. (It may slow to a stop, but that is okay as long as it doesn't reverse directions).
   
3. Affix the spark tape on the air track. There is a groove for this purpose.
   
4. Place a metal bar under one end of the track.
   
5. Release the cart from that end of the track.
   
6. When the person releasing the cart is clear of the track, press and hold the red button to start the sparks. At the same time, click the stopwatch to start the time.
   
7. Release the red button before you get to the end of the track. At the same time, click on the stopwatch to stop measuring the time. You do not want to include any collisions or bouncing in your data.
   
8. Record the position of the cart as a functon of time in your data. (Important: the position is measured from the first dot; it is not a measurement of the position between dots).
   
9. Plot a graph of position versus time in GA. (Do not leave before you complete this step! You cannot do this kind of fitting on Excel or other spreadsheet program.) See the instructions below to do a curve fit.
   
10. You should plug into the automatic curve fit dialog box: v*x + 0.5*a*x^2. (The y on your graph represents the distance s in the notes, the x on your graph represents time).
    

Instructions for GA (save this! You will do this many times):

• To change the axis titles: double click on the data column header originally marked "x" and "y". Enter the title in the dialog box that pops up. A default graph title will appear once you have done this.
• To include uncertainties in your data: double click on the data column header and enter the amount of uncertainty in the appropriate box. (Note: you have two choices for errors, constant error or percentage error. Since you will most likely want a constant error, be sure to uncheck the percentage option.).
• To enter data : type in the data boxes. The graph window will immediately start plotting your data.
• To do a linear regression (also known as a least squares fit): after entering some data, click and drag your mouse from the lower left hand corner of the graph to the upper right hand corner. Make sure that you have included all your data points. Click on the button labeled "R=" in the tool bar above the graph.
• To do an arbitrary function curve fit: Click on the graph window to select it. Select "Automatic curve fit" from the menu. Type in your function or select a function from the list. Click "OK". A new window will pop up, and you will see boxes at the bottom with numbers that change rapidly. When the fitting is done, the numbers will stop changing. Select the fit if you are satisfied that the fit matches your data, if not, either select Try New Fit if you think you entered the formula wrong, or Pause and Aid Fit to try some new initial values.
• To add error bars to your graph: double click the graph and check the error bars option from the dialog.
• To erase connecting lines: double click the graph and uncheck the connecting lines from the dialog.
• To print your graph : click on the graph window to select it. Go to File>Print>Selected Display.
  

Data: (you may not need all of these rows)
height you have raised the track (measure the metal bar with a caliper): _________________+/-__________
length of the track: _______________________+/- ___________
 

Time (          )	Position from first dot (          ) +/- ______
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4

Assignments:

Calcuations

Show your calculation of .
Show your calculation of g.
Show your graph of position versus time. Include error bars, units, titles, and no connecting lines.

 Analysis

Justify your choice of error in the position, height, and length.
Calculate the error in g.
Compare g to the theoretical g.

Questions assigned:
1. Why might the experimental value of g be different from the theoretical one? What was the limiting factor in the accuracy of our experiment?






2. Do you think that the multiburst photo method described in the notes would provide a more accurate measurement of g? Why or why not?