| Table of Contents |
Crummett and Western, Physics: Models and Applications,
Sec. 8-3,4
Halliday, Resnick, and Walker, Fundamentals of Physics (5th
ed.), Chapter 8
Tipler, Physics for Scientists and Engineers (3rd ed.), Sec.
6-6,7
Figure 1 shows
a very simple mechanical system which we are going to examine from the
point of view of transformations and conservation of mechanical energy.
An object of mass M rests of a tabletop; it is connected by a light
string which passes over a pulley and attaches to a second mass m,
which hangs free. We will assume that friction at the contact between M
and the tabletop, and in the axle of the pulley, is so small that we can
neglect it.
As mass M moves from left to right in the diagram, mass m is rising, increasing its gravitational potential energy; and x is increasing. M does not move vertically, so it does not come into the potential energy calculation. Thus the potential energy of this system is
(1)
(U always contains an arbitrary constant.) x0 is any reference point I might choose; x = x0 is the point at which we define potential energy to be zero. If the string connecting the mass does not stretch, the two masses are both moving at speed v, and
(2)
If friction in the system is negligible, the sum of potential and kinetic energy is a constant of the motion:
(3)
As the mass M moves from left to right, U is increasing, so K must be decreasing, and the motion is slowing. At some point all the system's kinetic energy has been turned into potential energy: the system comes instantaneously to rest, then turns around and starts transforming U back into K -- that is, it moves back to the left with increasing speed.
On the other hand, if there is appreciable friction in the system, it will do negative (always negative!) Work on the system, and the total mechanical energy E will decrease monotonically as the motion goes on. The frictional force f, if any, can be considered constant, and
(4)
In the laboratory, you'll set up a situation closely equivalent to that of Figure 1. An air track approximates a frictionless horizontal surface, with a glider moving along it for mass M. Hanging mass m is connected over to it by a thread that passes over a pulley. The goals of the experiment are to see if Equation (3), conservation of total mechanical energy, is borne out in this system; and to investigate the frictional forces in the system, if frictional losses are observable.
The translation of Figure 1 into the laboratory is sketched in Figure 2 above. Note: Before beginning this experiment, you should already have completed the orientation exercise on the Sonic Ranger and the PC which appears at the beginning of this chapter.
(1) In this laboratory you will again be recording the position of an air-track glider using the sonic ranger. Position the sonic ranger a little beyond the end of the air track that is nearest the wall. Put a glider on the air track, turn on the blower (air output at position 3), and collect data with the SR. Adjust the SR until it can reliably "see" the glider all the way to the pulley end.) If necessary, level the track as best you can, by adjusting the screws in the feet of the track.
(2) After the track is level, turn off the air supply. Record the ambient temperature (there is a thermometer on the wall of the laboratory) and calibrate the sonic ranger.
(3) You are now ready to collect data for motion of the system of Figure 1. Attach a thread to the hanger, run the thread over the pulley at the end of the track, and hang a small hanger on the end. Measure the mass of your glider (leave the string attached) using the triple-beam balance; and also measure the masses of any weights you will use for m. Record these values in your lab notebook. Your goal is to record the motion of the system with at least three different values of mass (m) hanging on the end of the string. The mass values can range anywhere from 5 g to 30 g or more, depending on what masses are available.
(4) Pick one of the mass values you are going to use. Hang the mass on the end of the string. Place the glider at the end of the track farthest from the SR. Start the data collection and then gently push the glider toward the SR. Check the collected data to insure that it is clean and looks reasonable. You should see a parabolic "U" shape, as in Figure 3.
If you didn't catch
the returning glider, but let it bounce from the bumper, you might see
some cusps where the glider rebounds In this case, don't use data that
occurred after the glider rebounds. Oscillations occur in the pulley support
when the glider bounces; this is particularly noticeable when heavier masses
are used.
(5) When you have data you consider satisfactory save the selected data to disk. Be sure to use a different name for each file you store! Record the file name in your lab notebook, along with the mass hanging on the end of the glider and all other pertinent information.
(6) Fit a parabola to the x vs. t data. While you are still running the kinematics software, taking data, you can use the fitting tools to pass a trial-and-error parabola through the data. Remember not to look at the curve parameters until you've finished tinkering! Record the equation of the curve.
The program will also provide you with a velocity vs. time graph. What it does is simply (delta x)/(delta t) from the position data:
(5)
(Notice that the velocity values are internal to the software and are not saved when you save your data to a file.) During the time interval during which the x-t graph is parabolic, the v-t graph should be a straight line.
(7) Repeat the acceleration measurements of (4)-(6) above for at least two other mass values m on the end of the string.
The goal here is to check whether the sum of potential and kinetic energies does stay constant! Do the following for each of your data sets:
(1) Import the data set into Excel. Use Equation (5) to construct a column of velocity data corresponding to the position data. Construct three more columns containing the potential energy, kinetic energy, and total energy at each time value. (In computing U, notice Figure 2: what the program records is d, the distance from the flag to the ranger; x increases when d decreases.)
(2) Make
a graph showing the displacement of the glider vs. time; and another
graph of potential energy, kinetic energy, and the total mechanical energy
vs. time. Include these graphs for (two graphs for each of your
data sets) in your notebook at the appropriate place. (Graphs from the
data shown in Figure 3 appear at right.)
(3) Do your graphs support the idea that mechanical energy is a constant of the motion? Comment, explain, and/or discuss.
(4) If there is a small, steady decrease in your total-energy graph, it is probably due to frictional losses. If you observe this in your data, try to estimate the frictional force (f) from Equation (5). (If you do observe a measurable frictional force), does it appear to be more or less the same in all your data sets, or does it tend to increase with increasing m? This might tell you where (glider or pulley) the friction is acting.
If you didn't get your track very level in Procedure (1), the vertical motion of M might also cause a small secular change in E. How could you discriminate between this, and a frictional force?
DLH/GCK 5/96 - updated 7/97