What is a Solid Angle and How is it Calculated?

[j04015]

Homework Statement

I don't get how the units in v = ω r match up. v / r= ω and let's say v is in m/s and r is in meters. This would make ω in 1/s; where does the rad/s come from?

Relevant Equations

v = ω r

m/s / m = 1/s

 

[kuruman]

Radians refers to a dimensionless quantity used to specify the measure of an angle. If ##s## is the length of an arc on a circle of radius ##r##, the angle ##\theta## subtended by the arc is ##\theta = s/r##. The angle is dimensionless, so ω has units of inverse time.

 

[Mister T]

This is a very common quandary. The radian is a unit, not a dimension. The radian can just pop in or out of expressions, and it's of no consequence.

Consider an example like a car going around a circular track. The circumference of the track is 10 meters and the car travels at a speed of 2 m/s. How much time does it take for the car to make one lap?

You could write ##\frac{10 \mathrm{m/lap}}{2 \mathrm{m/s}}=5\ \mathrm{s/lap}##.

Or, you could write ##\frac{10 \mathrm{m}}{2 \mathrm{m/s}}=5\ \mathrm{s}##.

The lap is a unit, but it's not a dimension, so it can pop in and out of expressions with no consequences.

In other words, 1 rad/s is equivalent to 1 s^-1.

All dimensions are units, but not all units are dimensions.

 

[jack action]

$$\frac{\frac{m}{s}}{m} = \frac{\left(\frac{m}{m}\right)}{s}=\frac{rad}{s}$$
Here ##\left(\frac{m}{m}\right) \equiv rad## only because the measured lengths represent a radius and an arc length of the same path. It cannot apply to every length divided by another length.

 

[Mister T]

$$\frac{\frac{m}{s}}{m} = \frac{\left(\frac{m}{m}\right)}{s}=\frac{rad}{s}$$
Here ##\left(\frac{m}{m}\right) \equiv rad## only because the measured lengths represent a radius and an arc length of the same path. It cannot apply to every length divided by another length.

But note that ##\left(\frac{m}{m}\right) =1## in all circumstances. So the use of the radian is just a convention.

 

[kuruman]

See here for an accessible in-depth analysis of angles, their dimensions and more.

 

[vanhees71]

One should never use rad as a unit. An angle is a dimensionless quantity, being defined as the ratio of the arc length of the corresponding sector of a circle divided by its radius. Sometimes one writes "rad" to make sure that one indicates to use this definition of the measure of an angle.

There are other measures for the angle, of course, e.g., "degrees", where a right angle (i.e., the angle ##\pi/4##) is given the value ##90^{\circ}## or "gons" where the right angle is ##100 \;\text{g}##.

 

[haruspex]

One should never use rad as a unit.

I cannot think how else to use it.
"The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure"
https://en.wikipedia.org/wiki/Radian

 

[vanhees71]

I'd never write ##\phi=42 \text{rad}## but rather ##\phi=42##, because rad is just a factor of 1. There is no unit in a dimensionless number. There's a lot of confusion, using the symbol rad, as this discussion shows. As the Wiki article correctly states: ##1 \text{rad}=1##!

 

[haruspex]

There is no unit in a dimensionless number.

Then what is a degree (°)? I see no difficulty in having a unit that is dimensionless,

 

[vanhees71]

A degree is also an oddity ;-)). We have ##180^{\circ}=\pi \text{rad}=\pi##. So a degree is also just a number, ##1^{\circ}=\frac{\pi}{180}##. I once helped out as a tutor in a lab, and at this time one usually worked with pocket calculators. Many students couldn't get correct sine and cosine values, because they didn't know that you had to make sure, in which mode the pocket calculator was adjusted. Interestingly besides rad and degrees it also had gons :-). Again ##1 \;\text{gon}=\pi/200##.

 

[kuruman]

I'd never write ##\phi=42 \text{rad}## but rather ##\phi=42##, because rad is just a factor of 1.

Units and dimensions are two different concepts. It is a good exercise in clarity always to attach explicit units to a number. Saying "I bought 6" is less informative than "I bought 6 pounds" which less informative than "I bought 6 pounds of potatoes." Here the unit that ought to be specified is "pounds of potatoes." Even "pure" numbers have units, e.g. 1 gross = 12 dozen = 144 ones, "one" being the ultimate generic unit.

You can always write ##\phi=42 ##, but will that be unambiguous to the reader? I think that clarity must take precedence over correctness even at the risk of redundancy.

 

[gmax137]

Not to mention slope or road grade given in percent, "Trucks Use Low Gear, 6% Grade Next 3 Miles." A grade of 100% is 45 degrees (pi/4).

Do the road signs in SI countries say "...slope EDIT 0.05 radians ahead ..."

 

[kuruman]

Do the road signs in SI countries say "...slope EDIT 0.05 radians ahead ..."

No they don't. However road signs don't display units for anything in SI countries or anywhere else I've been. See speed limit example below. With road signs brevity of recognition takes precedence over clarity of units.

 

[Mister T]

There is no unit in a dimensionless number.

There is no dimension in a dimensionless number, but there certainly can be units for dimensionless numbers.

There's a lot of confusion, using the symbol rad, as this discussion shows.

True. It's been discussed thoroughly in the literature. It confuses most students because it's just a convention.

A degree is also an oddity ;-)).

I don't see what you mean by oddity. It's something we see and use a lot. It never seems odd to me when I use it or see it.

There are 12 eggs per dozen. If we eliminate the units because they are dimensionless, that becomes odd. Suppose I sent someone to the grocery store to get a dozen eggs but instead told them to go to the store and get me 12. Now that would be odd. I would need to specify, or at least make clear by the context, that I mean 12 eggs.

According to one dictionary oddity means "the quality of being strange or peculiar".

 

[kuruman]

##\dots~##but instead told them to go to the store and get me 12. Now that would be odd.

Sorry, 12 is even.

 

[jack action]

##\phi=42##

I'm confused, is that an angle or a solid angle? If only there was a way to clearly identify the difference:

The steradian is a dimensionless unit, the quotient of the area subtended and the square of its distance from the centre. Both the numerator and denominator of this ratio have dimension length squared (i.e. L²/L² = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different kind, such as the radian (a ratio of quantities of dimension length), so the symbol "sr" is used to indicate a solid angle.

 

[vanhees71]

We are discussing angles, not solid angles!

 

[haruspex]

We are discussing angles, not solid angles!

I think @jack action's point is that the use of rad or rad² tells you which. However, the discussion @kuruman linked in post #6 suggests plane angles and solid angles should have the same angular dimension. I suspect this is related to the way the exponent of ##\pi## in areas and volumes of spheres steps up every second dimension: 0, 1, 1, 2, 2, …##.

 

[vanhees71]

The "unit" for the solid angle is ##\text{sr}## (steradians) and not ##\text{rad}^2##. Of course all these "units" are simply 1!

 

[Mister T]

Of course all these "units" are simply 1!

Again, I don't know what you mean. Are you saying that one degree is simply 1?

And one radian is simply 1. That would confuse a student like the OP. One could conclude that since they are both equal to 1 they are equal to each other!

 

[jack action]

Again, I don't know what you mean. Are you saying that one degree is simply 1?

So a degree is also just a number, ##1^{\circ}=\frac{\pi}{180}##.

Of course all these "units" [ ##\text{sr}## (steradians), ##\text{rad}^2## ] are simply 1!

 

[vanhees71]

Yes. An angle is a ratio of lengths (arclength of a section of a circle divided by the radius), a solid angle the ratio of an area with the square of the radius of the corresponding radius of the cone:

https://en.wikipedia.org/wiki/Solid_angle