8.1.1 Balancing a Pool Cue

The center of mass of an object is the point at which the object balances in a gravitational field, no matter how the object is oriented. Another description, one that relates more directly to the integral, is that the center of mass is the average location of the mass distributed throughout the object. In general, whenever the problem is to find an average of something distributed in a continuous manner, the solution is likely to involve an integral.

Note 1 – Center of mass demonstrations

We'll start by considering the center of mass of a simple object, a pool cue - basically, a tapered wooden rod. A typical pool cue is \(58\) inches long with circular cross sections that taper from a diameter of \(1\;\;3/8\) inches at the butt to \(9/16\) inches at the tip (see Figure 1).

Now the balance point of a pool cue has a considerable effect on the "feel" of the cue. Our particular cue is weighted so that the center of mass is \(17\) inches from the larger end. How much does the weighting matter? Where would the center of mass be if the cue were shaped from a uniform piece of wood without the added weighting?

The fact that mass is distributed continuously along the length of the cue is what leads to the need for integration to solve this problem. Here is our strategy:

• We will work out how to find a center of mass in a discrete problem, such as hanging masses on a balance beam.

• We will approximate our continuous distribution of mass by a collection of discrete masses.

• We will improve the discrete approximation by using more, but smaller, masses.

If we choose our approximating terms appropriately, then as the discrete distribution of masses approaches the continuous one, the discrete average (a sum) will approach the appropriate continuous average (an integral).