3.2 Squares and Square Roots

We mentioned that the A/B scales can be used for multiplication/division, as can the C/D scales. But the primary use of the A/B scales is to quickly find squares and square roots. For a given length along the rule, the A scale covers a numerical range that is twice the numerical range on the D scale. Hence, we see that “log A” = 2 \(\times\) “log D” which tells us that A = D\(^2\).

The A scale gives the squares of the numbers found on the D scale; very convenient when one wants to calculate areas of squares or circles, calculate kinetic energies, calculate power given an electrical current and resistance, etc. In reverse, given a number on the A scale its “square root” will be found directly on the D scale. The same relationship of course holds between the B and C scales as for the A and D scales. The figure below shows a close-up of the B and C scales of our example slide rule. It is easy to spot on the B scale the squares of the numbers on the C scale. Is it easy to read off the value of \(\sqrt{2}\)?

Squares and Roots.

Suppose we wish to calculate the kinetic energy \(W\), in Joules, of an object of mass \(m\) = 2.7 kg moving at a speed \(v\) = 6.8 m/s. The formula for kinetic energy is \(W = \frac12 mv^2\). Let’s do the calculation two different ways. First, we can use the C/D scales to multiply all the factors together:

  • Line up the cursor to 5 on the D scale.
  • Align 10 on the D scale with the cursor, and move the cursor to 2.7 on the C scale.
  • Align the 1 on the D scale with the cursor, and move the cursor to 6.8 on the C scale.
  • Align the 10 on the D scale with the cursor, and move the cursor to 6.8 on C again.
  • Find the result at the cursor on the D scale: 6.25.
  • Estimate the answer: a bit less than 3/2 \(\times\) 49 \(\approx\) 75.
  • So, final answer is 62.5 Joules.

The above calculation took three settings of the slide. Alternatively, we can calculate the result with a single setting of the slide (in this particular case) utilizing the A scale. Start by placing the cursor on 2.7 on the A scale. Then, move the slide to line up the 2 on the B scale with the cursor. The result of dividing 2.7 by 2 will be found on the A scale, aligned with the 1 on the B scale. But note that the square root of this result is also found directly below on the D scale. Since the 1 on the C scale is also aligned with this result, we can now multiply by 6.8 by moving the cursor to 6.8 on the C scale. At this point we have on the D scale – under the cursor – the result of \(\sqrt{m/2}\times v\). The square of this number will be directly above on the A scale. It should read roughly \(\frac12 mv^2\) = 62.5 (Joules) and is our final answer.

Yet again, via computer:

## [1] 1/2 * 2.7 * 6.8^2 =  62.424 J

Notice, too, that the kinetic energy for other speeds as well, from 1 m/s up to about 8.6 m/s, can be read without having to move the slide.

\(~\)

Another common calculation is that of the length of the hypotenuse \(c\) of a right triangle given the lengths of the other two sides, \(a\) and \(b\), where \(c^2 = a^2 + b^2\). Suppose two sides have lengths \(a\) = 3.7 and \(b\) = 5.2 and we want to calculate \(c\). One way is to use the A scale and find the square of \(a\) (13.7) and the square of \(b\) (27), and – writing down the results if necessary – add the two results together (40.7). Then, move the cursor to this result on the A scale and find its square root directly below on the D scale: about 6.35.

Another approach is to rewrite the relationship as \[ c = \sqrt{a^2 + b^2} = a\sqrt{1+(b/a)^2} \] where we note that \(b>a\) for the computation that follows. This time, using just two settings of the slide and without having to write down any intermediate results, we can find the hypotenuse. Here are the steps to perform on the slide rule:

  • Move the cursor to 5.2 on the D scale.
  • Align 3.7 on the C scale to the cursor, thus dividing 5.2 by 3.7 (= \(b/a\)).
  • Sliding the cursor to the 1 on the C scale, we follow the cursor to the A scale to read the square of this result: 1.98 (= \((b/a)^2\)).
  • Mentally adding 1 to the result, we move the cursor to 2.98 on the A scale, \(1+(b/a)^2\).
  • The square root of the present result will be directly below on the D scale, \(\sqrt{1+(b/a)^2}\); so next we move the slide to align the 1 on the C scale with the cursor.
  • Moving the cursor to 3.7 on the C scale we are multiplying the previous result by 3.7 (\(a\)).
  • The final answer, \(c=a\sqrt{1+(b/a)^2}\), will be at the cursor on the D scale: \(c\) = 6.38.

Check:

## [1] sqrt( 3.7^2 + 5.2^2 ) =  6.38200595424354