The International System of Units — Question 7

Question 7

Unit one does not mean quantity without meaning

For a circle, define plane angle by $\theta=s_{\mathrm{arc}}/r$, where $s_{\mathrm{arc}}$ is arc length and $r$ is radius; the coherent SI angle unit is the radian, $\mathrm{rad}$. A full circle has arc length $2\pi r$.

A rotor completes exactly three uniform revolutions in $2.00\ \mathrm{s}$. Define cycle frequency $f$ as completed revolutions divided by elapsed time and angular speed $\omega$ as angle swept divided by elapsed time. One hertz is one inverse second.

1. Show why plane angle has dimension one. Does this make it physically interchangeable with every other ratio of lengths, such as a material’s fractional extension?

2. Find $f$ and $\omega$, retaining the units that best communicate their meanings. Explain why the same base-unit dimension does not force the same numerical value.

3. A display removes all quantity labels and shows “$1.50\ \mathrm{s^{-1}}$”. Can the reader determine whether this is cycle frequency or angular speed? Suggest a clearer report.

Question 7 - Solution

Unit one does not mean quantity without meaning

(a) Cancellation does not erase the kind of quantity.

$$[\theta]=\frac{[s]}{[r]}=1,\qquad\theta_{\mathrm{turn}}=\frac{2\pi r}{r}=2\pi\ \mathrm{rad}.$$

The ratio of two lengths has dimension $\mathsf{L/L}=1$. The radian is a special name for the unit one used for plane angle. A fractional extension also has dimension one, but it describes a different physical quantity. A dimensional match alone does not license identifying or adding unrelated quantities.

(b) Count revolutions and radians separately.

$$\omega=(2\pi\ \mathrm{rad})f=(2\pi\ \mathrm{rad})(1.50\ \mathrm{s^{-1}})=3\pi\ \mathrm{rad\,s^{-1}}.$$

One full revolution sweeps $2\pi\ \mathrm{rad}$. Therefore $$f=\frac{3}{2.00\ \mathrm{s}}=\boxed{1.50\ \mathrm{Hz}},\qquad \omega=\frac{3(2\pi\ \mathrm{rad})}{2.00\ \mathrm{s}} =\boxed{3\pi\ \mathrm{rad\,s^{-1}}}\approx9.42\ \mathrm{rad\,s^{-1}}.$$ Both reduce to inverse-time dimension, but one measures cycles per time and the other angle per time. Their numerical values differ by $2\pi$ when these units are used. Dimensional analysis cannot recover this geometric factor.

(c) Retain the quantity name.

No. The unit $\mathrm{s^{-1}}$ by itself does not specify which quantity the display reports. A clear report is “cycle frequency $f=1.50\ \mathrm{Hz}$” or “angular speed $\omega=3\pi\ \mathrm{rad\,s^{-1}}$”. Hertz is used for periodic frequency; retaining $\mathrm{rad\,s^{-1}}$ for angular speed makes the distinction visible.