What is the Unit of Angular Acceleration in SI Units?

[AakashPandita]

i solved a problem in all SI units and got certain value of angular acceleration. What is the unit of the value? Is it in radian/s^2 or what?

 

[DennisN]

Yes, it should be rad/s². But if you specify which equation(s) you used, it will be easier for others to analyze.

 

[AakashPandita]

But in the answer it was only 25/s^2

 

[hilbert2]

People often omit mentioning the unit of radians. For example, $\sin^{-1}\frac{1}{2}=\frac{\pi}{6}$ and not $\frac{\pi}{6}rad$.

 

[dauto]

But in the answer it was only 25/s^2

That's because radian isn't a physical unit. Angles are unitless physical quantities. radians are just an (optional) label attached to angles for convenience.

 

[Pythagorean]

That's not true; radians are physical units (same as degrees) and when you non-dimensionalize a system of equations, you have to account for that.

 

[The_Duck]

Angles measured in radians really should be thought of as dimensionless: for example, they can be thought of as the dimensionless ratio between the length of a circular arc and the circle's radius.

 

[dauto]

That's not true; radians are physical units (same as degrees) and when you non-dimensionalize a system of equations, you have to account for that.

That's not true. Look for instance at the equation that relates angular speed with tangential speed for an object in uniform circular motion: v=ωr, where v is measured in - say - m/s, w is measured in rad/s, and r is measured in m. Compare the units and you will see that radians are in fact dimensionless. The word radian is a reminder of what particular definition for an angle is being used, but it is not a physical unit.

 

[ModusPwnd]

The way I internalize that a radian is not a physical unit is to simply recall its definition. Its a ratio of the angle's arc to the total circle's arc. Like all ratios, the units cancel and you get a dimensionless quantity.

edit - doh, duck said this exact thing... yet somehow I missed it.

 

[Pythagorean]

That's not true. Look for instance at the equation that relates angular speed with tangential speed for an object in uniform circular motion: v=ωr, where v is measured in - say - m/s, w is measured in rad/s, and r is measured in m. Compare the units and you will see that radians are in fact dimensionless. The word radian is a reminder of what particular definition for an angle is being used, but it is not a physical unit.

It's a matter of convention that physicists ignore the radian in that case. Angular velocity technically has units of rad/s though. It is especially apparent when nondimensionalizing a system like the pendulum, where you treat the angle like any other dimensional quantity (breaking it into its dimensional constant and a non-dimensional variable).

It becomes even more important when talking about steradians in three-dimensional systems. If radians weren't a physical unit, then neither would be steradians, and now you're implying that volume has the same dimensions as area.

 

[Pythagorean]

oops, didn't mean to imply dimensional as above. Just that it's a physically meaningful unit.

 

[dauto]

It's a matter of convention that physicists ignore the radian in that case. Angular velocity technically has units of rad/s though. It is especially apparent when nondimensionalizing a system like the pendulum, where you treat the angle like any other dimensional quantity (breaking it into its dimensional constant and a non-dimensional variable).

What do you mean a convention to ignore the radians? The radians show up in one side of the equation but not the other. That mismatch of units is a no-no for dimensional physical quantities. How do you explain that except for the fact that angles are dimensionless quantities?

It becomes even more important when talking about steradians in three-dimensional systems. If radians weren't a physical unit, then neither would be steradians,

That's right. Steradians are also dimensionless

and now you're implying that volume has the same dimensions as area.

How so? I don't see it.

 

[Philip Wood]

I used to think that a radian wasn't a proper unit because an angle in radians is one length (arc length) divided by another (radius). But then it struck me: how do we measure mass? We might collide the body of unknown mass explosively with a body of known mass 1 kg (both at rest initially), and find the inverse ratio of the velocities after the bodies have separated. This will give us the unknown body's mass in kg. Is this process so very different? [A genuine question]

 

[Pythagorean]

The radians show up in one side of the equation but not the other.

They're on both sides, we just ignore them in two places. Remember that an arc is actually (r)(theta). The tangential velocity, then, would be (rad)(m/s). So you have

V = wr

rad*(m/s) = rad*(m/s)

This all dropped in the canonical physics discussions of tangential velocity though, since we a priori describe it in the context of a circle (r being the radius, v describing motion around the perimeter of the circle).

Also, I guess we have to be careful about what we mean when we say dimension vs. physical unit. They are used interchangeably sometimes. All I'm challenging is your assertion that "radian isn't a physical unit". Not the rigorous definition of dimension because radians indeed have no "covering properties", only a directionality. But direction is important in physics (otherwise we wouldn't need vectors)!

 

[dauto]

They're on both sides, we just ignore them in two places. Remember that an arc is actually (r)(theta). The tangential velocity, then, would be (rad)(m/s). So you have

V = wr

rad*(m/s) = rad*(m/s)

This all dropped in the canonical physics discussions of tangential velocity though, since we a priori describe it in the context of a circle (r being the radius, v describing motion around the perimeter of the circle).

All fine and dandy except that speed is actually measure in m/s. You can't simply drop a dimensional unit. That would be wrong. But we do drop rads from the units all the time. How come? Hint: That's because they are not dimensional units.

Also, I guess we have to be careful about what we mean when we say dimension vs. physical unit. They are used interchangeably sometimes. All I'm challenging is your assertion that "radian isn't a physical unit". Not the rigorous definition of dimension because radians indeed have no "covering properties", only a directionality. But direction is important in physics (otherwise we wouldn't need vectors)!

Now you're arguing semantics. The question was about why the rads were missing from an answer. The answer is that rads are not dimensional so they are not required to match. My personal private definition for "physical unit" is that it's a synonym for dimensional unit.

 

[dauto]

I used to think that a radian wasn't a proper unit because an angle in radians is one length (arc length) divided by another (radius). But then it struck me: how do we measure mass? We might collide the body of unknown mass explosively with a body of known mass 1 kg (both at rest initially), and find the inverse ratio of the velocities after the bodies have separated. This will give us the unknown body's mass in kg. Is this process so very different? [A genuine question]

In this example you're not measuring the mass. You're actually measuring the ratio between two masses which is indeed adimensional.

 

[Pythagorean]

All fine and dandy except that speed is actually measure in m/s. You can't simply drop a dimensional unit. That would be wrong. But we do drop rads from the units all the time. How come? Hint: That's because they are not dimensional units.

No, you can't just drop a unit, you have to justify it. And it is justified in the very reduced problem within which your point is valid, it's just not justified explicitly for undergraduate students.

If you want to describe velocity in general though, you need direction. The deal-breaker though, is that angular frequency and frequency are not equivalent... they differ by the unit of radians. Hz (1/s) are not rad/s.

I think the OP's book was careless, personally.

 

[dauto]

To me it sounds like you're dancing around the point without ever touching it. You still haven't explained how it is possible to drop a unit except for the fact that the unit is dimensionless.

 

[atyy]

dauto is of course right. You can treat 8 m as 8 times a standard metre. In a ratio of lengths metre/metre is 1. If a quantity is multiplied by 1, we can always leave out the multiplication by 1.

 

[Pythagorean]

Sorry, I thought it was pretty straightforward. It's because the question of tangential speed doesn't ask about direction or phase. Consider an arc. If you want to ask only what the length of the arc is, then you only need meters. But if you want to know how the arc curves in space, you want to know meter-radians. Does that make sense?

More specific to the spirit of the OP:
Do you understand why Hz (1/s) are not (rad/s) and why you can't drop rads in that case?

 

[Pythagorean]

dauto is of ciurse right. You can treat 8 m as 8 times a standard metre. In a ratio of lengths metre/metre is 1. If a quantity is multiplied by 1, we can always leave out the multiplication by 1.

I don't disagree with that but it's an inappropriate analogy. If you try to equate (1/s) to (rad/s), you're saying that 2*pi = 1.

 

[Philip Wood]

In this example you're not measuring the mass. You're actually measuring the ratio between two masses which is indeed adimensional.

How else do you measure mass? Aren't you - ultimately - measuring a ratio?

 

[Pythagorean]

How else do you measure mass? Aren't you - ultimately - measuring a ratio?

In some cases, I agree. And yet, we don't cancel out the units... we call the ratio of like units something entirely new. That sounds familiar. I think there's another unit that is so important that we don't want to lose the information about what ratios we were taking in the first place, so we give it a new name... Oh yeah, radians!

In other cases... mass can be expressed in units of length ^_^

 

[atyy]

I don't disagree with that but it's an inappropriate analogy. If you try to equate (1/s) to (rad/s), you're saying that 2*pi = 1.

1 cycle per second is 2 half cycles per second is 4 quarter cycles per second, but when you write that a frequency is 1 per second, you are presumably not implying 1=2=4=...

How else do you measure mass? Aren't you - ultimately - measuring a ratio?

When we write 1 kg, we mean 1 times the standard kg.

There is a corresponding ratio which is dimensionless, so you can take your pick of how to report it. If you report the ratio, then you would not multiply the final numerical answer by kg.

 

[atyy]

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).

 

[AlephZero]

Regardless of the learned discussions about ratios, if somebody tells me "the angle is 42", without any context, my first question would be "is that 42 radians, or 42 degrees, or 42 grads, or 42 complete revolutions?"

They are all "dimensonless", but not all the same size.

But then I'm an engineer not a physicist - but I remember we once hired a physicist to do some computer programming, and he couldn't understand why engineers wanted to input angles in degrees measured clockwise from the vertical, instead of in radians measured anticlockwise from the horizontal

 

[Philip Wood]

AZ: I like it! As a matter of interest, what units do engineers give $\omega$ in the context of reactances?

 

[dauto]

Regardless of the learned discussions about ratios, if somebody tells me "the angle is 42", without any context, my first question would be "is that 42 radians, or 42 degrees, or 42 grads, or 42 complete revolutions?"

They are all "dimensonless", but not all the same size.

But then I'm an engineer not a physicist - but I remember we once hired a physicist to do some computer programming, and he couldn't understand why engineers wanted to input angles in degrees measured clockwise from the vertical, instead of in radians measured anticlockwise from the horizontal

Yes, you're absolutely right. Those units are all dimensionless and they are all of different sizes. Those units represent numerical factors. That's different than a unit like the meter which represents an actual physical distance, not just a numerical factor. It is in that sense that radians are not a physical unit. Maybe an unrelated example would help.

If I tell you I had a beer last night you will probably picture something in your mind very different that if I said I had a dozen beers last night. Dozen and one are different sizes but they are both dimensionless. They represent numeric factors. So if I have a box of beers with a dozen beers in it, I can write the equation

1 Box = 12 bottles.

The unit box and the unit bottle are different sizes and they are both dimensionless. They are not physical units. It is often convenient to keep writing the word box or bottle as a reminder of what is it exactly we are counting, boxes or bottles. But when writing an equation, it is OK to drop those words if no confusion is possible. The word box is not a factor in the equation, it is just a reminder of what you're talking about.

Exactly the same thing is true about the equation

180 degrees = π radians

The units degree and radians are different sizes and they are both dimensionless They are not physical units. It is often convenient to keep writing the word radian or degree as a reminder of what is it exactly we are counting, radians or degrees. But when writing an equation, it is OK to drop those words if no confusion is possible. The word radian is not a factor in the equation, it is just a reminder of what you're talking about.

Units like meter or seconds are different. they are physical units. They are in fact factors in the equations. If you drop them, you are dropping a factor from the equation and the equation will be incorrect.

I hope the difference is now clear.

 

[atyy]

BTW, I hope it is well known that as the unit for frequency is Hertz, that for angular frequency is Avis

(I saw this in More Random Walks in Science.)

 

[Pythagorean]

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.

 

[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?