# What are the dimensions of angular velocity?

My friend said that angular velocity has dimensions of $$T^{-1}$$. Or equivalently, it's measured in $$\text{rad}/\text{s}$$, and $$\text{rad}$$ is dimensionless, leaving only the $$1/\text{s}$$.

But I think that angular velocity should have the same units $$L \, T^{-1}$$ as translational velocity, because both of them are velocities. Shouldn't the angular velocity be the distance traveled along the circumference per unit time? How could the dimensions differ?

• Your are mistaken. One way of giving the angular velocity is $\omega=\frac{|v| \sin{\theta}}{|r|}$ which gives $\frac{\frac{L}{T}}{L}=\frac{1}{T}$. We are talking about a change in angle over time, the spatial dimension is given by the $r$ radial distance from the origin without which the angular velocity has no meaning. – Feyre May 14 '16 at 10:35
• @Feyre that should probably be posted as an answer – David Z May 14 '16 at 10:40

It is $T^{-1}$. Consider a rod of length $l$, marked at $l/4$, $l/2$ and $3l/4$, and let it rotate with angular velocity $\omega$ about the centre ($l/2$) point. Now quite clearly the end points are moving twice as fast -- they cover twice the distance per unit time -- as the points marked $l/4$ and $3l/4$, so the dimensions can not be $L/T$, as the whole rod has the same angular velocity. In fact the dimensions are $\mathrm{angle}/T$, but angle, being the ratio of two lengths (diameter and circumference), is dimensionless, giving dimensions of $T^{-1}$.

The physical angular velocity is angular displacement divided by time interval. The dimension of angular displacement is angle (A); the dimension of time interval is time (T). So the dimension of angular velocity is A/T.

Confusingly, what is commonly known as "angle" (e.g. in the SI) is the radian measure of the angle: if theta is an angle (dimension A), the corresponding SI "angle" is its radian measure, "theta" = theta/rad (dimension A/A = 1). The SI "radian" is the radian measure of 1 radian: "rad" = (1 rad)/rad = 1.

The radian measure of theta, theta/rad, is the number of radians in theta--i.e. a number (dimension 1). So what is called "angular velocity" (in the SI and almost everywhere else) is the number of radians swept out (dimension 1) divided by the time interval (dimension T): its dimension is 1/T. In other words, (SI) "angular velocity" is the actual physical angular velocity divided by one radian.

As an example, if theta = 20 pi rad (dimension A) and t = 10 s, the (actual physical) angular velocity is omega = (20 pi rad)/(10 s) = 2 pi rad/s. The SI "angular velocity is: "omega" = omega/rad = 2 pi (rad/rad)/s = 2 pi 1/s = 2 pi "rad"/s. [The SI helpfully explains that 1 can be replaced by (the SI) "rad" where appropriate (the quotation marks are not used).]

The well-known widespread confusion permeating this subject stems from the ubiquitous formula:

"theta" = (s/r)

where "theta" is said to be "measured in radians." What this actually means is that "theta" is the numerical value of theta when theta is expressed in radians: theta/rad--i.e. the radian measure of theta. So the formula is actually:

This is dimensionally consistent: dim(LHS) = A/A = 1; dim(RHS) = L/L = 1.

Rearranging for theta:

Your are mistaken. One way of giving the angular velocity is $\omega=\frac{|v| \sin{\theta}}{|r|}$ which gives $\frac{\frac{L}{T}}{L}=\frac{1}{T}$. We are talking about a change in angle over time, the spatial dimension is given by the $r$ radial distance from the origin without which the angular velocity has no meaning.