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 m₁g is the weight of mass m₁ (pull of the Earth) and the force m₂g is the weight of mass m₂. In these diagrams, we are assuming that m₁ > m₂, so that m₁g > m₂g. Thus, we expect m₁ to accelerate downward and m₂ to have an equal acceleration upward.

From the left-hand free-body diagram:

F_net = T - m₁g = m₁a

The right-hand diagram yields the following equation:

F_net = m₂g - T = m₂a

Adding the two equtions to eliminate T yields:

m₂g - m₁g = m₂a + m₁a

Solving for the acceleration of gravity results in:

m₂ - m₁
a = ------------ g
m₂ + m₁

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.

1. 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.

2. To this, add a circular, hanging reflector to one of the masses, we'll call this mass m₁. 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 m₂. 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 m₂ is at the pulley and m₁ 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.

3. 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, m₁ 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, m₂ is too heavy. If it accelerates all the way to the floor, m₁ is too heavy.

4. 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 m₁ another push. The goal here is to get m₁ to drop and m₂ 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.

5. 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 m₁ 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.

6. 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, m₁ and m₂, 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.

1. 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 m₂ to counter most of the mass of the reflector. Simply removing that mass will produce the desired mass differential, giving masses of approximately m₁ = 220 g and m₂ = 200 g, though not precisely. Be sure to use the precise values for your apparatus.

   Figure 5. Graphs With Acceleration.

2. 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 m₂ 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.

3. 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 m₁ began to move toward the Sonic Ranger. Using the Data/Analyze Data A analysis tool, we can find that starting point.

4. 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 (h₁) and time (t₁) at that point.

5. 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 (h₂) and time (t₂) at that point. You now have the total distance, h = h₂ - h₁, and total time, t = t₂ - t₁, for the Atwood's machine in free motion.

6. 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:

1. Using the equation for constant acceleration, h = (1/2) a t², find the acceleration of the system.

2. Using the expression in the discussion section, find the acceleration due to gravity. Remember to use the agreed upon values for m₁ and m₂, which ignores the frictional mass of the paper clips and a portion of the reflector.

QUESTIONS:

1. 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?

2. 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?

3. When you perform each data run, why must the paper clips be left on the masses? (What is their combined weight doing for us?)

4. For each set of trials, what do you calculate the speed of the masses to be just before mass m₂ hits the pulley?

5. 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?