Slide Rule Polar Play
It’s kind’a cool today, so I thought I’d explore some more slide rule functions. Specifically, I may do some examples of converting numbers between polar and rectangular forms. This is handy for doing most anything involving not just mathematics and trigonometry, but application areas such as surveying, electronics and civil engineering, just to name a few. Numbers in rectangular form are also known as complex numbers, and the j symbol is also known as the \(\sqrt{-1}\). Try doing that on your calculator. A nice book covers that radical number, Nahin (1998).
Once again I will be using the Post Versalog 1460, one of the more widespread and available slide rules, mostly because it was so popular during the slide rule heyday. It has T, ST/Sec T, S/Cos scales for this purpose. Basically, we are determining the sides and/or angles of a triangle such as, \(\theta = \tan^{-1}(\frac{b}{a})\), \(c = \frac{a}{\cos{\theta}}\) and \(Z(\cos{\theta} + j \sin{\theta})\). We can also find c easily by Pythagoras’ Theorem,1 \(c = \sqrt{a^{2} + b^{2}}\). This is the form we are using for conversions,
\[a + jb \quad \Leftrightarrow \quad c\angle\theta\]
Rectangular to Polar Conversion
For conversion, remember this rule: If a is larger than b, read the black figures on the T scale; if b is larger than a, read the red figures on the T scale. If one or the other number is between 0.1 and 0.01, use the ST scale.
For our first simple example, we have 3 + j4.2 As b is larger, we set the index to 4 (D scale), slide the hairline (HL) to 3 (D scale). We read the red T scale and find 53.15o. Next we slide 53.15 of the Cos scale under the HL, and read c = 5 on the D scale, giving the polar form of \(5\angle53.15^{o}\).
In this example a is larger than b, 125 + j30. Set index at 125, slide HL to 30 (D scale), read black T scale for 13.5o. Move slide to place 13.5 (S scale) under HL, read 128.5 (D scale) at index, giving the polar form of \(128.5\angle13.5\).
Polar to Rectangular Conversion
For simplicity, we will now convert the above two examples back to rectangular form. This also allows to check for errors.
So, the first example, \(5\angle53.15^{o}\), we set the index to 5 (D scale), move the HL to 53.15o (Cos scale), read 3 (D scale), then move HL to 53.15 (S scale), and read 4 (D scale) for 3 + j4.
The second example, \(128.5\angle13.5^{o}\), we set the index to 128.5 (D scale), move the HL to 13.5 on Cos scale. This requires using the right index, for a reading of 125. To read the sine of 13.5o once again requires swapping indices, then moving the HL to 13.5o (S scale). This indicates 30 (a/b ratio > 10) on the D scale for 125 + j30. If the angle were 1.375o, we would set on the ST scale, and the D scale would indicate 125 + j3.
Soon I hope to perhaps post something using some K&E slide rules. Time will tell, but for now, that’s all for this endeavor. And it’s a new year, and not just any new year, but a Leap Year, which means February has an extra day. I went to High School with a guy who was born on the 29th of February. So he experienced only four birthdays by the time he attended High School!
Anyway, God Bless!