A Determination of g, the Acceleration Due to Gravity,

using Newton's Laws of Motion.

Introduction

Newton's second law says that the net force on an object is its mass times its acceleration.  For the setup sketched below, the pulling force is provided by the weight of mass m1, W1=m1g.  Although g varies with location, a good average value of g is 9.80 m/s2 for motion near the earth's surface.

In this experiment you will measure g (testing whether 9.80 m/s2 is a good value) using some known masses, an understanding of Newton's second law and a record of the motion.

Mass m2 is a cart, able to roll (with some friction) on your lab table. It is connected to the hanging mass m1 by a string passing over a pulley. When the mass is dropped, the change in position of m1 (and m2) is recorded on the tape every 1/10th or 1/60th of a second (depending on switch setting). The acceleration of m1 (and m2) will depend on the value of g. (If you could do this experiment on the moon, would you expect the acceleration to be greater or less than that on earth? Why?)

Theory

The individual forces pulling the masses are shown in Figure 2a.2. T is the tension in the string while m1 is moving downward. The string pulls up on m1 with a force equal to the tension T and pulls m2 to the right with the same tension.  The kinetic friction force f opposes the motion of m2. m1g is the weight of m1. (Note that the vertical forces on m2 have been omitted, as they will not be needed in this analysis.)

The net force on m1 is downward and equals m1g - T. The net force on m2 is to the right and equals T -­ f. Since the accelerations of m1 and m2 must have the same magnitude, we can designate this magnitude by the letter a. Newton's second law then gives:

m1g ­- T = m1a          for m1 and

T -­ f = m2a         for m2.

m1g -­ f = (ml + m2) a.

Solving for g,

Thus if you measure m1, m2, f, and a, you can determine g.

Objective

In the experiment you will determine the cart acceleration, a, and the friction force, f, for different known masses m1 and m2. You will then use these results to obtain g, the acceleration due to gravity.

Equipment

Cart, pulley with mounting clamps, string, timer, timing tape, weights and sand bags.

Procedure

Set up the equipment as shown in Figure 2a.1. Use about 1 m of string and about 1 m of timer tape and adjust for at least 65 cm travel of ml and the cart. Fasten the tape to the bottom of the cart with sticky tape.

You must observe the following precautions.

1. Be sure the spark tape is passing through the sparker in the correct direction and that the tape is free to slide through the sparker.

2. Be sure that the pulley height is adjusted so that the string from the pulley to the cart is horizontal.

3. Be sure that the pulley, cart, and timer are all in one line with the cart aimed at the pulley.

Record three tapes for different masses: m1 = 0.500 kg, m2 = cart; ml = 0.300 kg, m2 = cart and m1 = 1.000 kg, m2 = cart plus sandbag.  For each run you must determine the actual mass of m2. For each m2 you must also determine the friction (see below).

Getting tapes:

To take data, hold the cart in a steady position with m1, just below the pulley.  Take any slack out of the timing tape and then start the sparker. Wait an instant and then abruptly, but without jerking, release the cart.  As soon as you obtain each tape, write the values for ml and m2 on it, so you do not get the tapes mixed up later.  When you have obtained your tapes, look at the timer marks on each to see if they appear consistent with motion that increases in speed as time goes on.

Determining friction:

Since friction is not negligible in this experiment, you must measure the friction force, f.  To do so, use Newton's first law.  Hang a weight on the string that produces a force equal and opposite to the kinetic friction.  When you get the right weight, the cart will move at constant velocity, either 0 or moving. Try hanging a small mass m = 0.025 kg. Give the cart a nudge and observe the motion (let a tape pass through the sparker without the sparker being on). Increase or decrease m to get a constant velocity after you nudge the cart. You should be able to determine the needed mass to 2 or 3 grams by direct observation, but you can take a motion record to check. Be sure to determine the friction force for all values of m2 that you use. Because the friction force is the mass times g and you are to determine g in this experiment, you must replace f in Equation (2a.1) by mg and solve for g.  Can you do the algebra to derive equation 2a.2 from 2a.1 when you replace f by mg?  Do it to get kudos!

Data Analysis

You now have a position record for points separated by constant time intervals.  Record the distances and times in an Excel spreadsheet.  Then determine the average velocity for each time interval.  Plot the average velocity versus elapsed time.  Determine the slope of the line in the average velocity versus time plot.  This should be the acceleration.  Use the slope and intercept to create a fit to the data.  Show a plot with both data and fit to see how good the fit is.  In your write-up, explain why the slope of the average velocity versus time plot is the acceleration!

Make such a plot for each tape!

Determine the slope a for each of your graphs. Each value will be different. Calculate the experimental value of g for each of your runs using Equation 2a.2 and your values for a, m1, m2 and m.

Calculate the percentage difference between each of your results and the expected value of g, 980 cm/s2 (9.80 m/s2).

Conclusions

Now that you have results for g you are probably tempted to say something like: "The experiment worked, I got results within a few percent of what I expected" or "My results are bad, I don't know what is wrong." Neither of these statements is very illuminating. No measurement is exact and in the "working" world of experimental science where the result is not known beforehand, a scientist must attach an estimate of the experimental uncertainty to any measurement made. If this is not done, there is no way to know whether or not a theory is contradicted by the data or whether it agrees within the experimental error.  Do you think that your value of g is consistent with an actual value of 980 cm/s2 or not.  Why or why not?  How precise were your measurements?  How far from the expected result would be enough to convince you that the expected result is wrong?