8.1.5 Calculation of Moment

Now let's calculate the moment of the pool cue. We approach this calculation in the same way we approached the calculation of the volume. Again, we imagine the cue divided into \(n\) sections of width \(\Delta x = 58/n\). Again, we approximate each section by a cylinder of constant cross-sectional radius. Then we calculate the moment of each of these approximating cylinders and add these moments to approximate the moment of the whole cue.

Recall that the volume of the \(k\)th approximating cylinder is

Thus the mass of the \(k\)th approximating cylinder is \(\delta\) times this mass or

Just as we assumed that the cross-sectional radius has the constant value \(r(x_{k-1})\), we'll assume that all this mass is concentrated at the left endpoint \(x_{k-1}\) of the interval \([x_{k-1},x_k]\). Then the moment of the \(k\)th section is approximately

\(x_{k-1} \times\) the approximate mass of the \(k\)th section \(= x_{k-1} \delta \pi [r(x_{k-1})]^2 \Delta x\).

Now our approximation to the total moment is the sum of the point moments

As \(n\) becomes large, this sum approaches both the total moment of the cue and the integral

Checkpoint 6

Example 1

Calculate the center of mass coordinate \(\bar{x}\) for the unweighted pool cue.

Solution   Now that we have integral representations for both the mass and the moment of the cue, we find

Notice that the \(\delta\)'s in the numerator and denominator canceled - we did not need to know a specific value for \(\delta\). Thus, for a uniform wooden cue, the center of mass is located \(21.3\) inches from the butt. 

Recall that the real pool cue has a center of mass \(17\) inches from the butt. Apparently this is a more comfortable position for the cue to balance, and weights were placed in the butt end to make this shift in the balance point.