# Ambiguity in dimensional analysis

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1. Jan 19, 2016

### DaTario

Hi All,

My question is twofold and follows:
1) Why the dimension of torque is not Joule, as it is Newton times meter?
2) Why the derivative of the velocity with respect to the distance cannot be measured in Hertz?

Thank you all,
Best Regards,

DaTario

2. Jan 19, 2016

### bcrowell

Staff Emeritus
Torque does have units of joules, and torque also has units of newton-meters. Clear communication is the reason we avoid writing it with units of joules.

3. Jan 19, 2016

### stockzahn

@1: Although it seems that torque and energy has the same dimension, you can see the difference, if calculated with vectors:

T = F x l
W = F ⋅ s

In the torque-case the lever l is rectangular to the force - no work is done.
In the work-case the displacement s is parallel to the force - the torque is zero.

4. Jan 19, 2016

### Staff: Mentor

The answer to question 2 is for the same reason that Ben gave for question 1.

Chet

5. Jan 19, 2016

### mfig

I agree with the other replies: it is mostly for clarity that these units are named differently. But there also is a real difference. For example, energy is a scalar quantity, torque is not. There is no directionality expressed in saying the internal energy of 1 kg of steam is 2 Joules. However there is directionality expressed in saying you applied 2 Nm of toque to that nut. (Were you tightening or loosening the nut, for example?) Torque can also be seen to drag the hidden dimension of radians in from how it is defined. The units of torque can therefore be thought of as Joules per radian. (I personally have never liked the way radians and cycles are ignored as a explicit unit in physics, as I think it leads to confusions like these.)

A similar consideration applies to your second question. Hz is actually cycles per second, not merely 1/s, though the cycles are usually ignored in unit statements, like radians. Cycles and radians in the compound units used in physics should be kept in mind to avoid confusion, even though it is not taught that way, in my opinion.

Hope this helps!

Last edited: Jan 19, 2016
6. Jan 20, 2016

### DaTario

I would like to express gratitude to you all for the answers.

Very interesting, spamanon, the observation that torque can be thought as Joule per radian. In fact, the equation:
$\tau = I \alpha$
makes it clear.

This makes the two questions very similar in deed.
I was aware that, regarding the question of the torque, the geometrical nature of the length in the two definitions makes work and torque substantially different.

Thank you all again,

Best wishes,

DaTario

7. Jan 20, 2016

### DaTario

Just to put it clear, according to spamanon:
Torque is measured in Joules times rad and not Joules divided by rad.
By the way, is this notion well stabilished ?

8. Jan 20, 2016

### mfig

9. Jan 21, 2016

### DaTario

Ok, spamanon, but in the equation:
$\tau = I \alpha = I \,\frac{d^2 \theta}{dt^2}$
the dimension of torque seems to be $K\!g \, m^2 \, rad / s^2$

How did you get Joules per radian?

Best wishes,

DaTario

10. Jan 21, 2016

### DaTario

Perhaps I have found out how did you get it...
Considering the figure attached, we may say that $\tau = F \, R$.

Then, we may express the work, $\tau$, as:
$F\, R = \frac{F \, v} {\omega } = F \, \frac{ds}{dt} \, \frac{dt}{d\theta} = F \, \frac{ds}{d\theta}$

which shows that work may be expressed as Joules per radian.

Is this derivation, ok? It seems ok to me.
It is curious, nevertheless, the subtlety involving of this apparent dimensional conflict between $\tau = I \, \alpha$ and $\tau = F \, \frac{ds}{d\theta}$.

Best Regards,

DaTario

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Last edited: Jan 21, 2016
11. Jan 21, 2016

### mfig

There is no conflict. Recall I said that keeping these units hidden leads to confusion?
When talking about rotations, radius is not in meters, but is meters per radian. For instance, recall that the formula for arclength is s = rθ = (meters/radian)*radian = meters.

Thus the units of I are kg⋅m2/rad2. The units of angular acceleration are rad/s2. So the units of the product Iα are then kg⋅m2/(s2⋅rad), or Joules/radian.

Last edited: Jan 21, 2016
12. Jan 22, 2016

### Svein

But since radians are dimensionless (arc length/radius where both have dimension meter), you can put "radians" wherever you want in a dimensional analysis...

13. Jan 22, 2016

### mfig

I am not sure what the point is here. Are you saying we cannot think of tourque as Joules/radian? SI seems to disagree with you, if you see the link above.

14. Jan 22, 2016

### Svein

No. I am saying that dimensional analysis is no help here.

15. Jan 22, 2016

### DaTario

The notion that radius must be defined as meter per radian really solves the problem, or what I have called the apparent dimensional conflict. In fact, this notion was present since the begining in my derivation done in post #10.

Additionally, I would say that there is, in introductory physics books, a well stabilished formula for work which clearly presents torque as Joule per radian, namely:

$W = \int_{\theta_i}^{\theta_f} \tau \, d\theta$.

Regarding the Svein comment, I believe that despite the fact that radians are dimensionless quantity, we are not allowed to locate it wherever we want on the formalism. I think radian have its proper place in definitions and expressions, just as if it were waiting to be considered as a respected dimension value by some authority.
Best wishes,

DaTario

Last edited: Jan 22, 2016
16. Jan 22, 2016

### DaTario

Taking this reasoning a bit longer:

Thinking about the problem of determining a circular orbit's velocity of a massive particle around a planet, at a given distante of its center, it turns out that we must conclude that the constant G of gravitation must have radian in its unit.

G should be measured in N m^2 / (kg ^2 rad ^2)

as the unit of force, Newton, should be: Kg m rad / s^2.

Is it correct?

Best wishes,

DaTario

17. Jan 22, 2016

### mfig

This is mysterious to me. Clearly, it is a help! Re: Why are the units of torque Joules but also not Joules? My answer is because it is not Joules at all, rather it is Joules per radian. J/rad makes much more sense physically for a torque if you think about the gaining of energy as an object is accelerated angularly due to an applied torque. And it makes much more sense than saying, "Torque is N⋅m, but we are not going to call that a Joule, even though 1 N⋅m is 1 Joule."

I have been thinking of radius, when rotations or turning forces are analyzed, in terms of m/rad ever since my 2nd year as a physics major, all through graduate school and into professional life. It helps me understand many things that others find confusing (like the OP here), and I am not alone.

Here is a recent paper written by two NIST physicists, one a Nobel winner, arguing that radians should be treated as a unit in SI and not as a dimensionless number, much as I have been using by my own discovery. This should at least makes the case that there is some room for variety here.

Of course you are free to do what you want, as I am.

Last edited: Jan 22, 2016
18. Jan 22, 2016

### DaTario

I have also got interested in such questions since long. One of my concerns in dimensional analysis had to do with proposing (or at least suspecting) that the period of oscilation was not as an interval of time purely, but a rate, since its inverse is a rate, namely, the frequency. Thus, the period T should be a rate, with dimensions of time per cicle.