DISCUSSION:

As Galileo knew, when a heavy object is dropped, its acceleration is so great that measurements of its motion are difficult. The way he solved the problem was to "dilute" the effect of gravity by using balls that rolled down an inclined track. In this experiment, you will "dilute" gravity in a different way in a different way to get an approximate measure of the acceleration due to gravity. (You will also have a much better clock to time the motion than Galileo could have even dreamed of.)

Figure 1. An Atwood's Machine. |

The Atwood's machine consists of a simple pulley, with a string draped over it and weights (masses) hanging on either side, as indicated in Figure 1. Obviously, whichever mass is greater will tend to accelerate downward, pulling the smaller mass upward. Since the two masses are attached with a non-stretching string, they will undergo the same acceleration. The positive direction chosen follows the string around the pulley, as indicated in the figure.

To analyze the motion, is it common to make three idealizations. First, the
pulley is assumed to have no effect, except to change the direction of the
force exerted by the string. (It is often called a "massless" pulley.) Second,
the entire system is assumed to have no friction and, third, the string is
assumed to have no mass. If a "light" pulley and a "light" string are used,
and the pulley mount is fairly good, these idealizations are approximately,
but hardly exactly, fullfilled. You should keep this in mind as you perform
and analyze this experiment.

Figure 2. Freebody Diagrams. |

To help in analyzing the motion, we may use two "freebody diagrams," as shown
in Figure 2. In each of the diagrams, the mass is represented by a point and
all the forces acting on the mass are shown. The force T is the tension in
the string, and it must act in the same way on both masses (same string!).
The force m_{1}g is the weight of mass m_{1} (pull of the
Earth) and the force m_{2}g is the weight of mass m_{2}. In
these diagrams, we are assuming that m_{1} > m_{2}, so that
m_{1}g > m_{2}g. Thus, we expect m_{1} to
accelerate downward and m_{2} to have an equal acceleration upward.

From the left-hand free-body diagram:

The right-hand diagram yields the following equation:

Adding the two equtions to eliminate T yields:

Solving for the acceleration of gravity results in:

This is the experssion you will use to find the acceleration due to gravity approximately from your experimental data.

PURPOSE:

The main purpose of this experiment is to make an approximate measurement of the acceleration due to gravity, using methods that Galileo might have used, with computer aided enhancements. In addition, you will learn something about the difference between idealized and real apparatus.

APPARATUS:

Bar stand and clamp, pulley, string, set of masses, hanging reflector, Sonic Ranger, protective cage, and a Macintosh computer.

Figure 3. The Experimental Arrangement. |

EXPERIMENTAL PROCEDURE:

Part I. Finding the Frictional Force.

- Set up the apparatus as shown in Figure 3, using equal masses of about
200 g each. With the Sonic Ranger on the floor underneath any hanging
apparatus, keep the protective cage over it at all times. A simple string
break can send massive bits of apparatus plummetting into the active element,
ruining the device. The length of the string should be chosen so as to give
an approximate value of h, or the freedom of travel, of about 50 cm, or 0.5 m.
Taking into concideration the string that drapes over the pulley and the
string involved in the knots, about 110 cm should be a good working length.
The best way to tie two 200 g masses to opposite ends of a string draped over
a closed pulley is to thread the string through the pulley and tie onto the
first mass. Then hang that mass on the pulley's support structure, removing
its weight from the string as you tie the other end to the other mass.
- To this, add a circular, hanging reflector to one of the masses,
we'll call this mass m
_{1}. Since the reflector will have a definite mass (About 20g, although the exact figure will be written somewhere on it.) that mass will have to be approximately balanced out with an additional mass on m_{2}. Due to the relatively small size of the pulley and the large size of the reflector, the two masses cannot pass each other in free motion, effectively halving their possible freedom of travel, but since they can't pass each other, they can't collide during the experiment either, removing a possible source of error. You now need to take into concideration the minimum distance the reflector needs to remain from the Sonic Ranger. (Listed as h_{min}~ 40 cm in the figure.) By adjusting the masses so that m_{2}is at the pulley and m_{1}is at its lowest point of travel, it becomes possible to reposition the pulley support structure on the bar stand so as to maintain that approximately 40 cm minimum distance with the reflector. - At this point, the apparatus will not be in perfect equilibrium, but
should be close enough for us to proceed. Place the masses side-by-side and
release them. Since the mass of the reflector is likely to be greater than
the mass you used to approximately balance it, m
_{1}will very likely accelerate downward very slightly. If it doesn't, give it a*small*push downward to get it going. If it stops too quickly or even reverses direction, m_{2}is too heavy. If it accelerates all the way to the floor, m_{1}is too heavy. - If there is any acceleration (deceleration is just a negative
acceleration), add a single paper clip from the supplied box to the lighter
side, reposition the masses side-by-side and give m
_{1}another push. The goal here is to get m_{1}to drop and m_{2}to rise with a constant velocity (no acceleration). The reason being that the*real*pulley we are using contains a component of friction. Friction being a*force*that opposes motion, and a force containing*acceleration*, no acceleration means no friction, or to be more accurate, the forces of friction have been cancelled out by other forces. Repeat this step until you've convinced yourself that the masses aren't accelerating.Figure 4. Graphs Without Acceleration. - Turn on the Macintosh computer and launch the MacMotion program. Pull up
the usual three graph layout of Distance, Velocity, and Acceleration. Change
the upper limits on all of the time axices to 5 s, as the free motion of the
Atwood's machine doesn't take very long. Replace the masses side-by-side.
Click Start and wait about one second while the computer takes data before
giving m
_{1}a push. You should obtain a set of graphs that look like those in Figure 4.You should be able to to convince yourself whether or not you've succeeded in getting an acceleration of zero by looking at the regions of the graphs showing free motion. (From about 1.5 s to 3.5 s in the figure.) Is the acceleration graph hovering about 0? Is the velocity graph a horizontal line? Is the distance graph linear? If the answers to each of these questions (They're all asking the same thing.) are "yes" then the frictional forces have been balanced.

- Print out the set of graphs you believe best demonstrates the balance of
forces. At this point, it's instructive to figure what the value of the
frictional force is. Find the difference between the two masses, m
_{1}and m_{2}, as they stand in equilibrium. That difference in mass accounts for friction. We can concider both masses to have the value of the lower mass since the difference in weight is nullified by the frictional force.

Part II. Measuring the Acceleration Due to Gravity.

- We now want to change the the mass of one or both hanging masses to give
an effective difference in mass of 20 g. You probably have an extra 20 g mass
hanging on m
_{2}to counter most of the mass of the reflector. Simply removing that mass will produce the desired mass differential, giving masses of approximately m_{1}= 220 g and m_{2}= 200 g, though not precisely. Be sure to use the precise values for your apparatus.Figure 5. Graphs With Acceleration. - Holding both masses side-by-side again, click Start, wait about one
second and release. With a mass differential, no push should be necessary.
You should now have a set of graphs that look like those at right in Figure
5. Don't worry about what's happening in your graphs after m
_{2}has risen to hit the pulley. (About 2.3 s in the figure.) That's not free motion. That's not data we're interested in. The free motion from about 1 s to 2.3 s is what we want to look at. - We now have the task of gleaning from these graphs the starting and
ending positions and times. We want to know how far the masses went and how
much time it took them to get there. We can stipulate that the starting point
will be that point where m
_{1}began to move toward the Sonic Ranger. Using the Data/Analyze Data A analysis tool, we can find that starting point. - In the lower right-hand corner are two radio buttons, one for the Tangent
function and one for the Integral function. Click on the Tangent button to
turn it on now. We can now use that tool to determine our starting point.
Concentrate on the point in the Distance vs. Time curve where the straight,
horizontal line first begins to arc downward. Somewhere in there, the slope
of the Distance vs. Time curve will begin to go negative from zero. We can
find the point where the difference between the slope at one point and the
slope at an adjacent point differs by about 0.01, or about 1/100 m/s. This
may, though not necessarily coincide with the first point where the tangent
line in the Distance graph changes to something other than horizntal.
Choose the latter point and record the distance (h
_{1}) and time (t_{1}) at that point. - We can now stipulate the ending point to be that point where the
deceleration (acceleration) stops and changes direction. In other words,
the peak of the acceleration spike which indicates the end of free motion.
Again, using the Tangent tool and concentrating on the Acceleration graph,
there will be two adjacent points where the tangent line switches over from
positive to negative. It's very unlikely you'll see a perfect slope of zero
at the peak of the acceleration curve. Choose the point with the greatest
acceleration and record the distance (h
_{2}) and time (t_{2}) at that point. You now have the total distance, h = h_{2}- h_{1}, and total time, t = t_{2}- t_{1}, for the Atwood's machine in free motion. - Repeat the entire procedure using a 40 g mass differential and, if you have time a 60 g differential, printing out at least one good set of graphs.

Analysis:

- Using the equation for constant acceleration, h = (1/2) a t
^{2}, find the acceleration of the system. - Using the expression in the discussion section, find the acceleration due
to gravity. Remember to use the agreed upon values for m
_{1}and m_{2}, which ignores the frictional mass of the paper clips and a portion of the reflector.

QUESTIONS:

- If the Atwood's machine were an ideal system (no friction, massless
pulley and string) and two equal masses were used, what would happen if the
one mass were given a small puch downward?
- When adding paper clips to one of the masses, why are you trying to get
the masses to move at a
*constant speed*when given a small push? - When you perform each data run, why must the paper clips be left on the
masses? (What is their combined weight doing for us?)
- For each set of trials, what do you calculate the speed of the masses to
be just before mass m
_{2}hits the pulley? - Do your calculations for g yield a number that is greater than or less then the accepted value? Does this seem reasonable? Why, or why not?

Experimental Procedure and all original figures are copyright © 1998 Cathy Garrett.

Purpose, Apparatus, and Questions are copyright © 1995 Philip DiLavore with alterations copyright © 1998 Cathy Garrett.

All html code is copyright © 1998 Cathy Garrett.