How is tangential velocity measured in m/s when it’s r * w?

[vhmth]

TL;DR

Trying to figure out how v=r*w equates to m/s.

I feel very dumb asking this question, but I can’t seem to find anyone else asking it. I understand geometrically how we get to: $v = r * \omega$

Where v is the tangential velocity, r is displacement (meters), and omega is angular velocity (radians/s). Isn’t this then m*rad/s? How do we get m/s? The best I can come up with is that we’re just assuming an infinitesimal delta-theta, so we just kinda ignore the “radians” part, but I’m not satisfied with that leap.

 

[robphy]

Possibly useful reading:

The radian—That troublesome unit
Gordon J. Aubrecht, II; Anthony P. French; Mario Iona; Daniel W. Welch; The AAPT Metric Education and SI Practices Committee
Phys. Teach. 31, 84–87 (1993)
https://doi.org/10.1119/1.2343667 (paywall)


Dimensionless units in the SI
Peter J Mohr and William D Phillips
Metrologia, Volume 52, Number 1
DOI 10.1088/0026-1394/52/1/40
https://iopscience.iop.org/article/10.1088/0026-1394/52/1/40 (open access)

On the dimension of angles and their units
Peter J Mohr, Eric L Shirley, William D Phillips and Michael Trott
Metrologia, Volume 59, Number 5
DOI 10.1088/1681-7575/ac7bc2
https://iopscience.iop.org/article/10.1088/1681-7575/ac7bc2/meta (open access)

 

[Orodruin]

Just like a newton is a derived SI unit that boils down to kg m s^-2, the radian is a derived SI unit that boils down to m m^-1 = 1, ie, dimensionless. It is used mainly to denote that the dimensionless number you are writing down refers to an angle.

 

[vhmth]

Possibly useful reading:

The radian—That troublesome unit
Gordon J. Aubrecht, II; Anthony P. French; Mario Iona; Daniel W. Welch; The AAPT Metric Education and SI Practices Committee
Phys. Teach. 31, 84–87 (1993)
https://doi.org/10.1119/1.2343667 (paywall)


Dimensionless units in the SI
Peter J Mohr and William D Phillips
Metrologia, Volume 52, Number 1
DOI 10.1088/0026-1394/52/1/40
https://iopscience.iop.org/article/10.1088/0026-1394/52/1/40 (open access)

On the dimension of angles and their units
Peter J Mohr, Eric L Shirley, William D Phillips and Michael Trott
Metrologia, Volume 59, Number 5
DOI 10.1088/1681-7575/ac7bc2
https://iopscience.iop.org/article/10.1088/1681-7575/ac7bc2/meta (open access)

Wow these links are priceless. Thank you so much! Loved this:

For example, in the current SI, it is stated that angles are dimensionless based on the definition that an angle in radians is arc length divided by radius, so the unit is surmised to be a derived unit of one, or a dimensionless unit. However, this reasoning is not valid, as indicated by the following example. An angle can also be defined as 'twice the area of the sector which the angle cuts off from a unit circle whose centre is at the vertex of the angle' [7]. This gives the same result for the numerical value of the angle as the definition quoted in the SI brochure, however by following similar reasoning, it suggests that angles have the dimension of length squared rather than being dimensionless. This illustrates that conclusions about the dimensions of quantities based on such reasoning are clearly nonsense.

Also, this is so clear:

In calculus and physics it is convenient to use radians or rad for angle units. The angle in radians between two lines that cross at a point is the length of circular arc s swept out between the lines by a radius vector of length r from the crossing point divided by the length of the radius vector.

So it seems like, yes, the answer is that the dimension is m / m (dimensionless) if you consider the angle derived from arc length / radius, but you can get the same angular quantities other ways as well. Other ways which lead you to having real dimensions. Fascinating!

 

[Orodruin]

Loved this:

Unfortunately that passage is just misleading. The unit circle has a radius of 1 (dimensionless) and therefore also a sector of it will have a dimensionless area. The corresponding definition would be in terms of the area sector divided by twice the radius squared - which is again dimensionless.

Edit: If you think about it, you could use the same argument to assign $\pi$ dimensions of length squared …

Edit 2: Fixed minor typo.

 

[DaveE]

How do we get m/s? The best I can come up with is that we’re just assuming an infinitesimal delta-theta, so we just kinda ignore the “radians” part, but I’m not satisfied with that leap.

At any instant in time velocity is a vector with a magnitude having dimensions of m/sec. The fact that you know that it's direction is about to change based on an applied force, shouldn't change the concept of velocity. Velocities often change.

OTOH, it certainly is convenient for us to study radial motion and apply other metrics to it.

Velocity and radial velocity are two different things, but a particle can have both at the same time. Both versions are correct in the right context.

 

[PeroK]

Wow these links are priceless. Thank you so much! Loved this:


For example, in the current SI, it is stated that angles are dimensionless based on the definition that an angle in radians is arc length divided by radius, so the unit is surmised to be a derived unit of one, or a dimensionless unit. However, this reasoning is not valid, as indicated by the following example. An angle can also be defined as 'twice the area of the sector which the angle cuts off from a unit circle whose centre is at the vertex of the angle' [7]. This gives the same result for the numerical value of the angle as the definition quoted in the SI brochure, however by following similar reasoning, it suggests that angles have the dimension of length squared rather than being dimensionless. This illustrates that conclusions about the dimensions of quantities based on such reasoning are clearly nonsense.

It's this that is nonsense. The arc length of a circlular sector of radius $R$ is:
$$l = R\theta$$And we see that $\theta$ is dimensionless. And its area is:
$$A = \frac \theta 2 R^2$$And, again, we see that $\theta$ is dimenionless.

Where the author goes wrong, therefore, must be when he/she replaces the general radius $R$ with a dimensionless unit radius. Then, the formulas become:
$$l = \theta, \ A =\frac \theta 2$$And this only makes sense if we recognise that for a dimensionaless unit circle, both the arc length and the area are themselves dimensionless. As @Orodruin pointed out.

 

[vhmth]

It's this that is nonsense. The arc length of a circlular sector of radius $R$ is:
$$l = R\theta$$And we see that $\theta$ is dimensionless. And its area is:
$$A = \frac \theta 2 R^2$$And, again, we see that $\theta$ is dimenionless.

Where the author goes wrong, therefore, must be when he/she replaces the general radius $R$ with a dimensionless unit radius. Then, the formulas become:
$$l = \theta, \ A =\frac \theta 2$$And this only makes sense if we recognise that for a dimensionaless unit circle, both the arc length and the area are themselves dimensionless. As @Orodruin pointed out.

I could be missing something fundamental to dimensional analysis (since it’s a field I know nothing about lol), but I don’t see how $\theta$ is dimensionless. I can kinda see it if we’re talking about theta by itself, but what about in relation to an area or arc length?

I think where I’m failing to follow is, if there are two separate dimensionless quantities $\theta$ and $R$, what does multiplying them even mean? Since we’re giving them meaning by equating to the arc length, shouldn’t we impose some dimensionality to make sense of its product? How can we make sense of things that are dimensionless but make up 2 dimensions (for lack of a better word) of the final outcome? How is it practical to speak about things in this manner?

 

[PeroK]

I could be missing something fundamental to dimensional analysis (since it’s a field I know nothing about lol), but I don’t see how $\theta$ is dimensionless. I can kinda see it if we’re talking about theta by itself, but what about in relation to an area or arc length?

I think where I’m failing to follow is, if there are two separate dimensionless quantities $\theta$ and $R$, what does multiplying them even mean? Since we’re giving them meaning by equating to the arc length, shouldn’t we impose some dimensionality to make sense of its product? How can we make sense of things that are dimensionless but make up 2 dimensions (for lack of a better word) of the final outcome? How is it practical to speak about things in this manner?

I've no idea what you are asking. Post #7 seems straightforward to me.

 

[PeroK]

Mathematically you can have something like:
$$x =\cos t$$But, physically, you must have:
$$x = x_0\cos(\omega t)$$Where $x$ and $x_0$ have units of length. And $\omega$ has units of inverse time (aka angular frequency).

The pure mathematical equation is dimensionless.

 

[Orodruin]

I could be missing something fundamental to dimensional analysis (since it’s a field I know nothing about lol),

You really should familiarize yourself with it. It is an incredibly powerful tool. I suggest you start here: https://www.physicsforums.com/insights/learn-the-basics-of-dimensional-analysis/
Read that before continuing reading below.

but I don’t see how $\theta$ is dimensionless. I can kinda see it if we’re talking about theta by itself, but what about in relation to an area or arc length?

I think where I’m failing to follow is, if there are two separate dimensionless quantities $\theta$ and $R$, what does multiplying them even mean? Since we’re giving them meaning by equating to the arc length, shouldn’t we impose some dimensionality to make sense of its product? How can we make sense of things that are dimensionless but make up 2 dimensions (for lack of a better word) of the final outcome? How is it practical to speak about things in this manner?

Let us take the relationship between angle $\theta$, radius $r$, and arc length $\ell$:
$$\ell = \theta r$$ where both $\ell$ and $r$ have dimension length ($\mathsf L$). Taking the physical dimension of both sides results in $$[\ell] = \mathsf L = [\theta r] = [\theta][r] = [\theta]\mathsf L.$$ Since the physical dimension of both sides must match, it follows directly that $$[\theta] = \mathsf L/\mathsf L = \mathsf 1$$ ie, $\theta$ is dimensionless.

The above is the dimensional analysis of the arc length relationship with radius and angle. I suggest you attempt the dimensional analysis of $A = \theta r^2/2$ yourself.

 

[Orodruin]

Mathematically you can have something like:
$$x =\cos t$$But, physically, you must have:
$$x = x_0\cos(\omega t)$$Where $x$ and $x_0$ have units of length. And $\omega$ has units of inverse time (aka angular frequency).

The pure mathematical equation is dimensionless.

Unless you are using dimensionless units for time of course, but that’s another subject entirely. Very useful for making generalized graphs and implementation into computer simulations.

Some wise person (cannot remember who - not me, not sure I qualify) is rumored to have said the main difference between mathematicians and theoretical physicists is that the theoretical physicists keep one base dimension…

 

[vhmth]

You really should familiarize yourself with it. It is an incredibly powerful tool. I suggest you start here: https://www.physicsforums.com/insights/learn-the-basics-of-dimensional-analysis/
Read that before continuing reading below.

Sorry for the delay. I wanted to make sure I fully digested this and understood it. Holy crap I can’t believe I’ve been learning physics on my own without people like you to help all along. Thank you so much. I’ve bookmarked this and still have to go over the Buckingham pi theorem a couple more times, but things are a lot more structured now. :)

Since the physical dimension of both sides must match, it follows directly that

This was very helpful. It has made me realize that I can logic my way through all of this stuff but still have no idea how to practically think about the “radians” part of angular velocity (“radians per second”) being dimensionless. I don’t actually even know what it means to not have dimension (compared to a meter or second) in the context of angular velocity since it still makes sense in my brain that something is spinning and there has to be some intrinsic dimension to that “spin” in the same way changing the direction of a vector should change its meaning.

Ok I’m going to revisit this article a few times and see if it will eventually stick. Thank you all!

 

[PeroK]

This was very helpful. It has made me realize that I can logic my way through all of this stuff but still have no idea how to practically think about the “radians” part of angular velocity (“radians per second”) being dimensionless.

Whenever you divide one quantity by a quantity with the same dimensions, you have a dimensionless quantity. The natural angle in radians is a ratio of length/length, hence dimensionless.

I don’t actually even know what it means to not have dimension (compared to a meter or second) in the context of angular velocity since it still makes sense in my brain that something is spinning and there has to be some intrinsic dimension to that “spin” in the same way changing the direction of a vector should change its meaning.

Suppose you take the ratio of the vertical component of velocity and the horizontal component of velocity. That defines the tangent of an angle. You can represent the velocity of a particle by:
$$\vec v = v_x \hat x + v_y \hat y = (v\cos \theta) \hat x + (v\sin \theta) \hat y$$Where $v = |\vec v|$. That equation has dimensions of $LT^{-1}$. The angle itself has no dimensions, as:
$$\theta = \arctan\big ( \frac {v_y}{v_x} \big )$$Where $\dfrac {v_y}{v_x}$, being the ratio of two velocity components, is clearly dimesionless.

 

[sophiecentaur]

but I don’t see how θ is dimensionless.

Perhaps if you try to give θ a dimension you might be convinced. i.e. what could be divided by what?

There is the other question about why not always use angles in degrees and angular velocity in rpm. A very practical reason is that using radians avoids acres of 2π's all over your page when there's calculus or. anything 'trigonometrical' involved.

 

[weirdoguy]

how to practically think about the “radians” part of angular velocity

It is there, because people wanted to have unit for angle even though it is dimensionless so has no unit. I would say it is somehow psychological thing. But you don't have to use it! I've seen problem-books where authors do not use radians. I use them (rads) because "that's how I've been raised"

Here's an example from a polish book that not everyone uses radians:

 

[sophiecentaur]

I would say it is somehow psychological thing.

Not psychological; practical. Take a big formula out of a book with lots of ω's and θ's and re-write it in terms of degrees and frequency. But I can appreciate that it may be a bit intimidating and a bit showyoffy but Physicists have egos like the rest of us.

 

[PeroK]

Physicists have egos like the rest of us.

I wonder whether "ego" is a dimensionless quantity?

 

[sophiecentaur]

I wonder whether "ego" is a dimensionless quantity?

The "id" which is the alternative latin for self could be written 1D which implies one dimension.

 

[sophiecentaur]

I'm truly sorry about that one, chaps.

 

[WuliDancer]

I remember that the unit of angular velocity ω is degrees per second, and after converting ° into radians, there is no unit, that is, 180° = π (π has no unit).

 

[A.T.]

It is there, because people wanted to have unit for angle even though it is dimensionless so has no unit. I would say it is somehow psychological thing. But you don't have to use it! I've seen problem-books where authors do not use radians. I use them (rads) because "that's how I've been raised"

Dimensionless ratios are often reported with some "unit" attached, to indicate how the ratio was computed (how something was normalized). For example, in biomechanics the forces within a body are often normalized by bodyweight and then reported with the unit BW, while moments can be normalized by bodyweight * bodyheight (BWHt).

In that sense, angles in radians are arc-lengths normalized by the radius.

 

[sophiecentaur]

Dimensionless ratios are often reported with some "unit" attached, to indicate how the ratio was computed

We are neglecting the fact that the proper comparison should be between the radian and the single rotation or turn. The degree is arbitrary and so is a clock face and the hundredth of a circle. You could never imagine an angle measurement in radians; pick up a protractor with radians marked on it, with π half way and 2π back at zero. Madness. How you divide the circle is entirely down to the government.

 

[Orodruin]

pick up a protractor with radians marked on it, with π half way and 2π back at zero. Madness.

Why madness? Sounds perfectly viable to me.

 

[A.T.]

You could never imagine an angle measurement in radians; pick up a protractor with radians marked on it, with π half way and 2π back at zero. Madness. How you divide the circle is entirely down to the government.

Because the use of π to express radians is idiotic: "Let's define angles as arc length to radius ratio, but then express them using the circumference to diameter ratio."

 

[sophiecentaur]

Why madness? Sounds perfectly viable to me.

How would you use / draw a scale on the protractor to include more than '360 degrees)? You would need a spiral scale/ Isn't that madness? I'm bringing the real world into this thread.

 

[Orodruin]

How would you use / draw a scale on the protractor to include more than '360 degrees)? You would need a spiral scale/ Isn't that madness? I'm bringing the real world into this thread.

What does this have to do with using radians over degrees or vice versa?

 

[sophiecentaur]

the use of π to express radians is idiotic:

π is a ratio which you cannot change. It's not something you can choose. π is π in the same way that e is e. Those constants are outside any number base or system. Using radians in mathematics is a bit arbitrary but it does reduce the complexity of many calculations in the same way that natural logarithms do. Logs to the base ten are convenient but it's only at the end of a chain of calculations in order to present answers - we understand decimal notation and dB and half lives but we also happen to describe time constants in terms of exponentials. It's all pragmatic and there's nothing idiotic about the system.
If you really want to tidy up our use of numbers to describe our world then first sort out international standards of units.

 

[sophiecentaur]

What does this have to do with using radians over degrees or vice versa?

They each have their place in the World it's always a matter of habit and convenience. People tend to struggle when they realise they should be prepared to be bi-lingual in these matters.

If you happen to be a theoretical Physicist then you might never to present results other than in radians. If you were an automotive Engineer (or any Engineer for that matter you might never use radians. Revs and degrees are extremely convenient in practical life. You could divide one revolution into any arbitrary number of course.

 

[A.T.]

π is a ratio which you cannot change.

You can use a different ratio. Since radians are defined as arc-length / radius, it would make much more sense to express them as fractions of circumference / radius, rather than circumference / diameter.

There is a reason, why radians and trigonometric functions are defined on a unit circle, that has a radius of 1, not a diameter of 1.