If you don't have clean dry sand handy, that isn't a problem. Granulated sugar or salt work equally well, and if you are reasonably careful to keep things clean you can even use the stuff after the experiment.

Be sure to note which you use for your experiment.

The experiment consists of using a measuring cup (1/4 cup size) or a tablespoon (an accurate tablespoon, not a tablespoon from a silverware set) to build a series of conical piles containing measured amounts of sand (or salt or sugar; I'm going to say 'sand' here, but you can use any of the three materials).

The pile will be built on the figure below. You can click on Circle Graph to obtain the figure by itself, which might be useful if you need to print it out.

The graph consists of a set of 15 concentric circles (depending on how your browser and printer are set up the circles might not be exactly circular; that won't be a big problem). The 5th, 10th and 15th circles are labeled in both directions along each axis, with the labels just outside the circles. This will help you determine the average diameter of each pile.

You will begin by laying the circle graph on a flat horizontal surface. You will carefully pour a level 1/4 cup, or 4 level tablespoons, of sand onto the circle graph, doing your best to pour the sand right over the center of the graph. Be sure the sand in the cup or tablespoon is level, which can be accomplished by filling a bit over level and using a straightedge (e.g., the straight back of a knife) to scrape off everything above the edges of the cup or tablespoon.  NOTE:  If your graph prints out too small to hold several quarter-cups of sand, just use tablespoons instead of quarter-cups.

The sand will approximately form a circular cone with its peak above a point near the center of the graph. We wish to determine the radius of the circle formed by the base of the cone.

Of course the cone won't be perfectly circular and it won't be centered all that precisely. However, if you look to see what circle the cone reaches along each axis, you can take an average and determine the approximate 'average diameter' of the base.

To see how this might be done, look at the figure below, which depicts a top view of a hypothetical pile of green gourmet sand.

Along the positive x axis the cone base reaches between the 7th and 8th circles, somewhat closer to the 8th. We therefore might estimate that the cone base reaches 7.7 units from the center.

Along the negative x axis the base again reaches between the 7th and 8th circles, perhaps a bit closer to the 8th then along the positive x axis. We might therefore estimate that the cone base reaches 7.8 units from the center along this axis.

Along the positive y axis the base again reaches between the 7th and the 8th circles, about as close to the 8th as along the negative x axis. So we estimate that along this axis the cone base reaches 7.8 units from center.

Along the negative y axis the base reaches just about exactly to the 9th circle, so we estimate a distance of 9.0 units from center along this axis.

Our estimated distances are therefore 7.7, 7.8, 7.8 and 9.0 units from center. Averaging these distances we obtain 8.1 units (rounded to two significant figures), which give us a reasonable estimate of the radius of the base.

The figure below depicts these estimates and the average.

We continue adding sand 1/4 level cup, or 4 level tablespoons at a time (unless you are using tablespoons because of a small graph). Each time we use the same process to estimate average radius. Continue until seven of eight data points have been obtained.

Then plot a graph of the number of 1/4 cups vs. the radius of the base.

Recall that for the 3 parameters of a quadratic model we required 3 curve points, while for the 2 parameters of a linear model we required only 2 curve points.

For the present model, there is only 1 parameter, so theoretically only 1 curve point will be necessary to obtain the model. However, 1 curve point doesn't leave much margin for error in estimating its coordinates. So we average the values of the parameter a obtained for each of three curve points.

Sketch a smooth curve passing through the origin (the (0,0) point of the graph) and fitting the data points as closely as possible.

Pick a point on your curve somewhere between x = base radius = 1 and x = 3. Substitute the x and y values into y = a x^3 and evaluate a.

Pick two more points on the curve, in such a way that the chosen points are spread out from one end of the curve to the other.

For each point, substitute the x and y values into y = a x^3 and evaluate a.

Average your values of a. Use this average value to obtain your function model

y = a x^3, or (quarter-cups of sand) = a * (cone radius) ^ 3.

For each cone radius evaluate the function to determine the predicted number of quarter-cups for that radius. Plot your function on the same graph as the data points.

Determine the average discrepancy of each of your data points from your power function.

Observe whether there is any systematic pattern to the deviations.

If a computer or graphing calculator with curve-fitting options is available:

Fit a p = 3 power function to your data. See how close your estimate was to this function, and evaluate the quality of the fit for the best-fit function.

Fit a p = 2.5 power function to your data and evaluate the quality of its fit.

Repeat with a p = 3.5 power function.

Determine from among p = 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.5 the value of p that yields the best model for your data.

If a computer or graphing calculator with appropriate options is not available, you may defer this part of the assignment for up to a week in order to allow you to use the computer labs on campus.

1. Suppose that we wanted to build sand cubes instead of cones. We would have to use cube-shaped molds and add a small amount of adhesive in order to hold the cubes together, but we could certainly do so if we were determined enough.

Suppose that a single quarter-cup of sand makes a cube 1.5 inches on a side. How many quarter-cups would be required to make a cube with twice the scale, 3 inches on a side? (Hint: imagine building a 3-inch cube out of 1.5-inch cubes. How many would be required? Remember that a 3-inch cube is 3 inches wide, 3 inches deep, and 3 inches high.)

What value of the parameter a, for the y = a x^3 model of y = quarter-cups vs. x = cube width, would model this situation (use the data point (1.5, 1) and evaluate a)?

How many quarter-cups does this model predict for a cube three inches on a side? How does this compare with your previous answer?

What would be the side measurement of a cube designed to hold 30 quarter-cups of sand?

2. Someone screwed up and used 1/2 cup instead of 1/4 cup. The best-fit function was y = .002 x^3. What function would have been obtained using 1/4 cup?

A pendulum is constructed by tying a string to a weight. The string should be thin and light (a length of thread works very well as long as the weight isn't too great), and about 8 feet long. The weight should be pretty symmetrical so you can determine with reasonable accuracy where its center is, and should be reasonably dense (a hollow plastic ball isn't dense enough). A round onion, a roughly circular potato, a golf ball or a toy rubber baseball (or a real one) would work well; a volleyball, a basketball or a beach ball would not work so well.

Begin by holding the pendulum string at a point one foot (about 30 centimeters) from the center of the weight. Start the pendulum swinging and count the number of complete swings in a minute. A complete swing starts from an extreme point in the pendulum's motion, where the pendulum stops and reverses direction, and returns to that point.

You will repeat this exercise for pendulum length of 2, 3, 4, 5, 6 and 7 feet. For the greater lengths you might have to tie the string to a short stick to hold the pendulum high enough; if so brace the stick to keep it as stable as possible. You will record data consisting of swings per minute vs. string length.

• If you don't have 7 feet of string, or if you don't have the means to support a 7 foot pendulum, you may use shorter lengths.  Two possible sets of lengths are

10, 20, 30, 40, 50, 60 and 70 centimeters

2, 4, 6, 8, 10, 12, and 14 inches.

If you use one of these alternatives, be sure to specify which one you used.

Now determine for each length the time in seconds required for a complete swing. Make a table of time per swing vs. pendulum length. Then follow the procedure of the preceding exercises to fit a p = .5 power function y = a x ^ .5 to the data, and evaluate the fit.

4. For your number of swings vs. length data, use DERIVE or a graphing calculator to obtain power function fits y = a x ^ p for p = -.3, -.4, -.5, -.6 and -.7, and plot the resulting functions with your data points. Which function fits best?

5. For your time per swing vs. length data, use DERIVE or a graphing calculator to obtain power function fits y = a x ^ p for p = .3, .4, .5, .6 and .7, and plot the resulting functions with your data points. Which function fits best?

6. If the number of swings per minute is n, then what expression represents the number of seconds required for one swing?

If the number of swings per minute is y, then what expression represents the number of seconds required for one swing?

If the number of swings per minute is a x ^ -.5, then what expression represents the number of seconds required for one swing? Using your value of parameter a from #4 (or the value of a determined from your original graph, if you are deferring your DERIVE work), what function do you predict for the time per swing vs. length? How does this prediction compare with the function you obtained for time per swing vs. length?

7. If the time per swing, in seconds, is T, then what expression represents the number of swings per minute?

If the time per swing in seconds is y, then what expression represents the number of swings per minute?

If the time per swing is a x ^ .5, for the value determined previously for the parameter a, then what expression represents the number of swings per minute? How does this expression compare with the function you obtained for the number of swings per minute vs. length?

7. Using your best model of number of swings per minute, determine the range of lengths that corresponds to frequencies of 20 - 40 swings per minute.

8. Using your best model of time per swing, determine the pendulum lengths that would result in periods of .1 second and 100 seconds. Describe how you would construct each pendulum.

9. If one pendulum has length x1 and the other has length x2, then the ratio of their lengths is x2 / x1. What expressions, in terms of x1 and x2, represent the frequencies (i.e., number of swings per minute) of the two pendulums?

What expression, in terms of x1 and x2, represents the ratio of the frequencies of the two pendulums?