1. Nov 11, 2013

### 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?

2. Nov 11, 2013

### DennisN

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

3. Nov 11, 2013

### AakashPandita

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

4. Nov 11, 2013

### hilbert2

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

5. Nov 11, 2013

### dauto

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.

6. Nov 11, 2013

### 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.

7. Nov 11, 2013

### 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.

8. Nov 11, 2013

### dauto

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.

9. Nov 11, 2013

### 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.

Last edited: Nov 11, 2013
10. Nov 11, 2013

### Pythagorean

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.

11. Nov 11, 2013

### Pythagorean

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

12. Nov 11, 2013

### dauto

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?
That's right. Steradians are also dimensionless
How so? I don't see it.

13. Nov 11, 2013

### 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]

14. Nov 11, 2013

### Pythagorean

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

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)!

15. Nov 11, 2013

### dauto

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.
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.

16. Nov 11, 2013

### dauto

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

17. Nov 11, 2013

### Pythagorean

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.

18. Nov 11, 2013

### 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.

19. Nov 11, 2013

### 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.

Last edited: Nov 11, 2013
20. Nov 11, 2013

### 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?

21. Nov 11, 2013

### Pythagorean

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.

22. Nov 11, 2013

### Philip Wood

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

23. Nov 11, 2013

### Pythagorean

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 ^_^

24. Nov 11, 2013

### atyy

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=...

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.

25. Nov 11, 2013

### 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).