What is the Unit of Angular Acceleration in SI Units?

[dauto]

If the NIST standard (see: The International System of Units: Physical Constants and Conversion Factors) for rad/s as a physical unit isn't enough, there's a more rigorous discourse here:

http://khimiya.org/volume14/radian_.pdf

But there's really nothing new there on top of what I've already said. It also highlights the important difference between "dimension" and "physical unit". If there's any confusion about the physical quantity being measured... it's called angle.

I read the article. I don't agree with its definition for rad. See for instance the second paragraph from the bottom of page 485 where it says

"sin(θ) clearly has no physical meaning. It should be remembered that θ and the unit rad are both physical quantities of the same kind. One must use the numerical value θ /rad as sin(θ / rad)".

If that were true than we would all have been very sloppy every time that we wrote the expression sin(θ) in our lives. That's a lot of sloppiness. Thankfully, it turns out that angles are adimensional quantities and the expression 1 rad = 1 isn't sloppy. True, angles are physical things but they happen to be adimensional physical things.

 

[dauto]

http://physics.nist.gov/cuu/Units/units.html

So looks like Pythagorean has a point. Radians is a way of multiplying by 1 (see Table 3). Indicating multiplication by 1 is optional but can aid in clarity (see Table 4).

I think your link shows that he doesn't have a point. 1 is not a unit.

 

[srijag]

The unit is indeed dimensionless. Radian is dimensionless. If you try to evaluate the dimensions of angular acceleration, it is T^-2. So the units of angular acceleration being s^-2 makes sense.

 

[Pythagorean]

If that were true than we would all have been very sloppy every time that we wrote the expression sin(θ) in our lives.

Context is important. Rad as the ratio is a different interpretation than rads in a solid angle. The first is a formal mathematical definition, the second is a useful physical unit describing a quantity of solid angle. There's nothing wrong with carrying the rad in your sin(θ), there's just no reason to in pure mathematics (and of course, you absolutely should outside of mathematics... like in physics and engineering... where people use degrees and Hz).

True, angles are physical things but they happen to be adimensional physical things.

And that's exactly what I said. Obviously, it has no covering property (dimensionality) in cartesian space. r=0 is a point at the origin no matter how you vary theta. But we're not really talking about covering properties from the context of the OP, are we? That would be a diversion.

The unit is indeed dimensionless. Radian is dimensionless. If you try to evaluate the dimensions of angular acceleration, it is T^-2. So the units of angular acceleration being s^-2 makes sense.

The problem is more obvious comparing frequency to angular frequency. One has units of 1/s, the other has units of rad/s... and it is explicitly defined as such by NIST. This has nothing to do with dimensionality, really: it has to with their being a factor of 2*pi difference between the two units:

1= (1 rev)/(2*pi rad)

addendum: Note also that the NIST also explicitly defines angular acceleration with rads (atty's link, table 4) so, the OP's book violated the national standard.

 

[Philip Wood]

It's interesting, isn't it? that we don't feel the need to give sines, cosines and tangents units, whereas we want to give units to angles, as measured by arc length/radius. Yet the trig ratios, and the angle, as measured by arc length/radius are all ratios of two lengths. If there were no other angle measure (for example, if nobody had ever thought of degrees), would anyone have wanted to stick a unit after an angle measured as arc length/radius?