Lab 3:  Newton’s Laws

The purpose of this experiment is to test whether Force = Mass x Acceleration.  The technique will be similar to that used to measure g in lab 2.  You will use the weight of a small mass, m, to provide the force that accelerates a system consisting of a large mass, M, the air cart + the aluminum “flag,” and the small mass, m.  If friction can be neglected,

(M + m) a  =  mg                                                           1.

or

a = (mg)/(M + m).                                                                                                2.

You will measure m, M and a to see if the measured acceleration really is given by the above equation, i.e. does ameasured = atheory where the theoretical value is given by the equation above, using the measured masses and g = 9.80 m/s2.

You will measure the acceleration, a, by recording the time the flag passes the sensors on the air track, the distance between those points and computing the velocity and then the acceleration of the cart.  The computer will record the times; you will measure the distances and compute the velocities and acceleration.

EXPERIMENT

How do you measure a?  You will measure a by measuring the position of the cart at different times and then calculating the average velocity as a function of time, just like in lab 2..  You will then use these average velocities to calculate ameasured.  There are four gates or sensors that tell when the flag on the cart passes them.  The computer will record the times when the flag enters and leaves the position of the sensor.  You need to make the measurements below, which will be your “raw data”.

1.       measure the positions of the cart (e.g. the front edge of the cart) when the flag enters and when it leaves the sensor position.  (Measure both the entering and leaving position for each sensor!)  If you have four sensors, you will measure eight positions.  Keep track of which positions go with each sensor or gate.

2.       Measure the mass of the cart (including the aluminum flag) and the small weights.  Also measure the mass of the tape.

3.       As a check, record a run without the small mass, i.e. no tape attached to the cart.  Start the data taking and gently shove the cart down the track.  If the track is level and frictionless, the velocity or speed should be constant (until it reaches the end).  If it isn’t, get the problem fixed before proceeding.

When you have recorded these you will attach a small weight (10g) to one end of the tape, and the other end of the tape to the cart.  The tape will hang over the air bearing with the small weight hanging down.  Hold the cart about 10 cm before the first gate (record this value), start the “Newton's Laws” program and then release the cart.  The computer will record and display the times the cart entered and left each gate or sensor.  You will need to supply a file name for each run.  Choose short descriptive names like Wt10g, Wt20g, Wt30g and Wt0g.

4.       Save the times for each gate.  (The computer program will do this for you.)  These times are accurate to about the nearest 1ms or 0.001s.

5.       Repeat for two more small weights ( 20g & 30g).

These represent your raw data.  Now you need to analyze them to find the measured accelerations for each small weight and compare it to the expected acceleration for that weight.

ANALYSIS

You will analyze the results using the spreadsheet. When you are done, take your floppy to a computer and start Excel. In Excel you can import the data into the spreadsheet by going to the File  header and clicking on it.  Then choose Open from the pull down menu, click on the Look in box and select the 31/2 floppy in drive A:.  You will then have to go to the bottom of the box to the box labeled Files of Type and select to look for text files.  Then select your first file to import by clicking on it and then clicking the Open button.  A “wizard” will guide you through the importing process.  In the first window, make sure you select the Delimited file form.  Then click Next.  In the second box select the comma delimited file form.  Click Next again and then finish.  Your file should have been imported into the spreadsheet.  Your imported data should look like that shown at the right.  Either cut this and paste it into an Excel file or save it as an Excel file, i.e. an .xls file type.  Do not leave it as a text file, i.e. a file of type .txt.

The gate column tells you which gate the cart entered or left.  Of course the time column gives the time in milliseconds.  I suggest you add 4 rows to the chart near the top (see below) so you can list the masses of the cart, weight and the tape.  I usually included one for g as well.  You will need to convert the times to seconds (column C) and add a column (D) of the position of the cart when it enters and leaves the gate.  Then calculate the velocities in the next column (E). The speeds in column E are average speeds and the times you use to calculate the acceleration should be those in the center of the time interval used to calculate the speed.  Put this mid-time in column F, so F12 will contain the average of C11 and C12.  If you had four gates you will have eight times and positions and can calculate seven velocities from these.

Make a plot of velocity vs. time (Use the times from column F and the velocities from E) and then calculate the “measured” acceleration by doing a linear regression of velocity vs. time.  This is probably your best estimate of the acceleration. Do a regression of the velocities in column E vs. the times in F.

The X variable coefficient is the slope or acceleration and the standard error of the coefficient is the “uncertainty” or standard deviation for the slope.  Compare this measured acceleration to the expected acceleration calculated from equation 2.  (When you do this calculation of the expected acceleration, you should consider the effect of the tape’s mass.  Where will it go?  Indicate whether you included it and if so, where!)  A sample sheet is shown below.  You may want to put each run on a separate worksheet.

In your discussion, consider how close the measured acceleration is to the expected acceleration and what factors might affect your results, e.g. your position measurements and friction.  Indicate HOW these factors would influence your results and try to be quantitative.  You should indicate the results of your “no small weight” run.  Was the acceleration zero for that run?  If not, how would that influence your results?

I’ve show a sample spreadsheet below.  Again, I’ve altered the data slightly.