What, if anything, is a square radian?

Question

My 1st year Mathematics BSc course notes on circular motion use \begin{align} \frac{d}{dt}(\sin\theta) &=\frac{d}{dt}(\sin(\omega t))\tag{1.1}\\ &=\omega\cos(\omega t),\tag{1.2}\\ \frac{d^2}{dt^2}(\sin\theta) &=\frac{d^2}{dt^2}(\sin(\omega t))\tag{2.1}\\ &=\frac{d}{dt}(\omega\cos(\omega t))\tag{2.2}\\ &=-\omega^2\cos(\omega t),\tag{2.3} \end{align} where

• $\omega$ is angular speed, a scalar, with units $\text{rad}\cdot s^{-1}$;
• $t$ is time, a scalar, with units $s$.

From this it seems to me to follow that $\omega^2$ has units $(\text{rad}\cdot s^{-1})^2=\text{rad}^2\cdot s^{-2}$. Is that right, and, if so, does a square radian have a physical meaning?

Answer 1

"rad(ians)" is just another word for the number $1$, a unit-less unit. An angle is some part of a full rotation at $2\pi$, a unitless number. However, to tell that any number stands for an angle and not some amount of apples or cake, there is the habit to add "radians" to it.

Answer 2

OP answering own question.

Remarks by Hagen von Eitzen and Lutz Lehmann have prompted me to think of the following answer.

A radian is a dimensionless fraction with value $$\frac{1}{2\pi}.$$

It seems to me to follow that a square radian is a dimensionless fraction with value $$\left(\frac{1}{2\pi}\right)^2=\frac{1}{4\pi^2}.$$

This can be thought of as the angle by which the angular speed increases each second.

Answer 3

While a square radian is just radians squared in the context of the question, I was taught that just as a radian is a unit of "linear" angle, so a squaradian or steradian is a unit of solid angle. The solid angle around a point is $4\pi$ steradians.