K-3 AIR TRACK EXPERIMENTS

1. PURPOSE:

To study accelerated motion of a glider on an air track.

2. APPARATUS:

Air track, spark recording tape, gliders of various masses, slotted weights, air pulley, weight hanger, length of 1/4 inch wide magnetic recording tape.

3. BACKGROUND:

(1) Simanek, Donald E. An Introduction to Data Analysis. Chapter 8.

(2) Experiment K-1 of this manual, for the method of data collection and analysis.

4. FLETCHER TROLLEY:

Fig. 1. Accelerating a glider on a level air track.

A glider moves with very little friction on an air track. If set in motion on a level track it therefore moves with nearly constant velocity. We can make the glider accelerate by applying a force on the glider. This will be done by attaching a length of mylar magnetic recording tape to the glider, passing the tape over an air pulley at one end of the track, and attaching small slotted weights to the free end.

The hanging weight must be small compared to the glider's weight. We want to keep the glider velocity very small at all times. The 50 gm weight hangers should not be used, for that much weight would produce too much acceleration. Attach slotted weights directly to the end of the mylar tape.

Take data for at least four different glider masses, and four different hanging weights. Conserve data tape by putting four records on each tape. Do this by slightly bending the spark wire on the glider to locate it at any one of four imaginary parallel "tracks" along the waxed tape. Therefore you will need only four lengths of tape for this part of the experiment.

5. TILTED TRACK:

Riser blocks of various thickness are available. They may be placed under the leg at one end of the track to provide a known, small, tilt angle. The glider will accelerate on the tilted track because the force of gravity on the glider has a component parallel to the track. No other accelerating force will be used in this part of the experiment.

Measure the glider acceleration for four different mass gliders, and in each case, for four different tilt angles. Record four tracks of data on each tape as in section 4.

The objective is to see how the acceleration is dependent on (1) the mass of the glider, and (2) the tilt angle.

6. ANALYSIS:

This is one of the "simpler" experiments you will perform in this course. Therefore take this opportunity to learn a basic technique of data analysis which is fairly standard, and has applicability to many of the more sophisticated experiments you may perform in the future.

In part 4 the object was to investigate the relation between glider acceleration and applied force, to see whether the data is in agreement with Newton's law, F = ma. The applied force F is very nearly equal to the weight of the hanging mass, F =~ W.

You took data for a number of different mass gliders, m. For each one you used several different weights, W, and measured the acceleration, a.

One approach to the analysis is strictly algebraic. For each case you use the data for W and m in W = ma to calculate the expected acceleration. Compare that with the experimentally measured acceleration in each case and note the discrepancy between theory and experiment. This is a relatively crude and unsophisticated approach which would seldom be used unless no better methods were available.

The formula W = ma is a linear relation between W and a, whose graph is a straight line.

Plot the experimental data, a vs. W, for a single glider. This line should have slope 1/m. Data from the other gliders give lines of different slopes. These may be plotted on one sheet of graph paper. This graph clearly shows the results, the quality of the data, and the agreement between theory and data.

One can also plot this data another way. If we plot all of the data for a single value of W, graphing a vs. 1/m, we should get a linear relation of slope W. As before, all of the data may be plotted on a single sheet, each value of W producing a different straight line. If you came to laboratory well prepared, you probably used the same set of weight values for each different glider, to facilitate making this graph. If you didn't, you can still extract data for a single weight from your previous graph.

Calculate the slope of each of the lines on the graphs. Now, on another piece of graph paper, plot these slopes against the corresponding mass m (or against the weight W = mg, if you like). Is this a straight line?

Similar methods may be used to analyze the data of part 5. The two objectives clearly suggest which graphs you should make.

6. QUESTIONS:

(1) Derive the formula for the acceleration of a glider on a tilted frictionless track, as a function of g and tilt angle. Make no approximations.

(2) Express the formula of question (1) in an approximate form valid for small tilt angles. (The angle θ will appear in the equation, but trig functions of the angle will not.)

(3) In the arrangement used in part 4, prove (derive) the fact that the tension in the mylar tape is T = m(g-a) where m is the hanging mass and a is the glider's acceleration.

(4) How small must the acceleration be for the tension in the mylar tape to be equal to the hanging weight, to 1% accuracy?

(5) In the light of questions (3) and (4) the force the tape exerts on the glider is the tension in the tape, so

T = Ma = m(g-a)

so the glider's acceleration is


     mg
a = —————
    M + m

where m is the hanging mass, and M is the glider mass.

The relation between a and m is therefore not strictly linear. Using your range of data values, how accurately would you have to measure a, m, and M to detect this non-linearity?

(6) In the light of questions (1) and (2), how accurately would you have to measure a and θ to conclusively distinguish between the exact and the approximate formulae?

(7) Look up coefficient of restitution in your textbook or other references. Can you calculate this quantity from your data? If so, what is it? If not, explain what you would need to do to determine its value.