1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ambiguity in dimensional analysis

  1. Jan 19, 2016 #1
    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,

  2. jcsd
  3. Jan 19, 2016 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
  4. Jan 19, 2016 #3


    User Avatar
    Gold Member

    @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.
  5. Jan 19, 2016 #4
    The answer to question 2 is for the same reason that Ben gave for question 1.

  6. Jan 19, 2016 #5
    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
  7. Jan 20, 2016 #6
    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,

  8. Jan 20, 2016 #7
    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 ?
  9. Jan 20, 2016 #8
  10. Jan 21, 2016 #9
    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 ##
    yielding Joules radian and not Joules per radian.

    How did you get Joules per radian?

    Best wishes,

  11. Jan 21, 2016 #10
    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,


    Attached Files:

    Last edited: Jan 21, 2016
  12. Jan 21, 2016 #11
    There is no conflict. Recall I said that keeping these units hidden leads to confusion? o_O
    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
  13. Jan 22, 2016 #12


    User Avatar
    Science Advisor

    But since radians are dimensionless (arc length/radius where both have dimension meter), you can put "radians" wherever you want in a dimensional analysis...
  14. Jan 22, 2016 #13
    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.
  15. Jan 22, 2016 #14


    User Avatar
    Science Advisor

    No. I am saying that dimensional analysis is no help here.
  16. Jan 22, 2016 #15
    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,

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

  18. Jan 22, 2016 #17
    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. :woot:
    Last edited: Jan 22, 2016
  19. Jan 22, 2016 #18
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook