47
$\begingroup$

Recently, I was doing my homework and I found out that Torque can be calculated using $\tau = rF$. This means the units of torque are Newton meters. Work & Energy are also measured in Newton meters which are Joules.

However, torque isn't a measure of energy. I am really confused as why it isn't measured in Joules.

$\endgroup$

migrated from math.stackexchange.com Sep 21 '12 at 0:06

This question came from our site for people studying math at any level and professionals in related fields.

  • $\begingroup$ Minor note: Torque is usually given by $rF \sin \theta$, not just $rF$, unless the angle is always $90$ degrees of course because $\sin 90 = 1$. $\endgroup$ – Joe Sep 20 '12 at 23:45
  • 11
    $\begingroup$ Torque is a vector; energy is not. They just happen to have the same units. $\endgroup$ – Christopher A. Wong Sep 20 '12 at 23:46
  • 8
    $\begingroup$ Maybe this is helpful: if we do work on something by rotating it, the amount of work is the product of torque and angular displacement. Angular displacement is measured in radians, which is unitless, so torque must have the same units as energy. $\endgroup$ – Pink Elephants Sep 20 '12 at 23:48
  • 4
    $\begingroup$ en.wikipedia.org/wiki/Torque#Units $\endgroup$ – sdcvvc Sep 20 '12 at 23:49
  • 1
    $\begingroup$ More on units of torque: physics.stackexchange.com/q/36079/2451 $\endgroup$ – Qmechanic Sep 21 '12 at 9:50

10 Answers 10

56
$\begingroup$

The units for torque, as you stated, are Newton-meters. Although this is algebraically the same units as Joules, Joules are generally not appropriate units for torque.

Why not? The simple answer is because

$$W = \vec F \cdot \vec d$$

where $W$ is the work done, $\vec F$ is the force, $\vec d$ is the displacement, and $\cdot$ indicates the dot product. However, torque on the other hand, is defined as the cross product of $\vec r$ and $\vec F$ where $\vec r$ is the radius and $\vec F$ is the force. Essentially, dot products return scalars and cross products return vectors.

If you think torque is measured in Joules, you might get confused and think it is energy, but it is not energy. It is a rotational analogy of a force.

Per the knowledge of my teachers and past professors, professionals working with this prefer the units for torque to remain $N \ m$ (Newton meters) to note the distinction between torque and energy.

Fun fact: alternative units for torque are Joules/radian, though not heavily used.

$\endgroup$
  • $\begingroup$ I'd like to draw attention to the second to last paragraph, "Per the knowledge..." That's the paragraph that actually has the answer -- people prefer the distinction between J and Nm simply to remind them of what kind of quantity they are dealing with. $\endgroup$ – Cort Ammon Mar 14 at 16:36
19
$\begingroup$

Torque is force at a distance. Work is force through a distance. Same unit dimensions, different measurements.

$\endgroup$
  • 3
    $\begingroup$ This is the best answer. $\endgroup$ – Sushant Gupta Sep 4 '17 at 18:02
  • $\begingroup$ physically different properties... one is a scalar and the other, a vector. $\endgroup$ – Narasimham Mar 3 at 19:20
  • $\begingroup$ @Narasimham - ok, torque component along a fixed axis then. $\endgroup$ – ja72 Mar 3 at 19:53
  • $\begingroup$ They are conceptually defined differently in physics.The scalar is not even the absolute value of the vector. $\endgroup$ – Narasimham Mar 3 at 20:08
  • $\begingroup$ @Narasimham - A torque vector encodes the location of the line of action (since it is a force at a distance), but the torque magnitude only encodes the perpendicular distance (moment arm). I can argue also, that work is not just a scalar, but the result of a dot product of force and displacement vectors, and torque the cross product of force and location vectors. So conceptually they are not that different, except for a different type of vector product used. $\endgroup$ – ja72 Mar 3 at 22:52
12
$\begingroup$

The reason we distinguish the two is that torque is vector quantity, where as energy is a scalar quantity. So while we give the magnitude of torque the same units as energy, there is in fact additional information that tells us the direction the torque is applied.

UPDATE: As dmckee has pointed out in the comments, to be perfectly corrected torque is a pseudovector, which is equivalent to a mathematical bivector in three dimensions. This distinguishes it from a true polar vector. The distinction is important since the dimension of the pseudovector is n-1 instead of n. This is important conceptually as it is critical to our understanding of conservative forces and central forces, and more specifically the conservation of angular momentum.

In particular, angular momentum conservation implies that motion under central forces will always be confined to a plane.

$\endgroup$
  • 6
    $\begingroup$ Strictly torque is a pseudo-vector, though we don't generally make that distinction in a introductory class. $\endgroup$ – dmckee Sep 21 '12 at 2:59
  • $\begingroup$ @dmckee Thanks! I updated based on this your comment because the dimensionality point is of particular interest at the moment, especially as it relates to conservative systems. $\endgroup$ – user11547 Sep 21 '12 at 11:49
  • $\begingroup$ Hi. Intriguing update! :) The link that you quoted seems to have been removed. Can you provide an alternative link in case you happen to know the availability of the same content elsewhere? Thanks. $\endgroup$ – Dvij Mankad Mar 27 at 1:24
8
$\begingroup$

Yes, torque has units of joules in SI. But it's more accurate and less misleading to call it joules per radian.

Let's take the simple case of a single force acting perpendicular to position (reference) vector: $$\tau=rF$$ To tease out energy from this equation, let's consider an infinitesimal change in (rotational) energy $dE$: $$dE=\tau\,d\theta=rF\,d\theta,$$ or $$\tau=\frac{dE}{d\theta}.$$ From this equation, one can interpret torque as the amount of rotational energy gained per radian of rotation. In other words, joules per radian in SI units. But, since one usually considers radians as unitless, this "simplifies" to just joules.

$\endgroup$
  • 1
    $\begingroup$ A radian is a dimensionless measure with a dimensionless unit of "Radians" (it is not unitless, just dimensionless, using the distinction here: en.wikipedia.org/wiki/Dimensionless_quantity). There are other measures of rotation that are dimensionless, such as "Turns" 1 Turn = 2pi Radians. When you measure an angle in Radians, you report the ratio of the arc to the radius. When you measure in Turns, you report the ratio of the arc to the circumference. Why couldn't it have happened that Torque = Joules/Turn? $\endgroup$ – Gus Apr 3 '15 at 2:39
7
$\begingroup$

Joule and Newton meter are two units that are algebraically identical; you might say they are two names for the same unit. This is not the only example: Ohms is a unit of resistance, while "ohms per square" is an algebraically identical unit of sheet resistance. Hertz is a unit of frequency, becquerel is a unit of frequency in the context of radioactivity. In Gaussian units there is a delightful example of five algebraically identical units.

Why do people use different names for the same unit? Just the simple reason: It facilitates communication and avoids misunderstandings. If I mumble something and point and say "50 newton meters", you can be pretty sure I'm talking about a torque; if I say "50 joules" you can be pretty sure I'm talking about an energy. Therefore, having these different terms helps reduce the frequency of communication mistakes (albeit only to a limited extent).

The fact that torque and energy have algebraically identical units does not mean torque and energy are the same; in fact, it means nothing whatsoever. Torque and energy are completely different concepts that just happen to have algebraically-identical units. (Well, I suppose torque and energy are connected in various ways, just as any two randomly-selected quantities in classical mechanics are connected in various ways.)

$\endgroup$
2
$\begingroup$

Torque could be measured in joules per radian. Torque by angle gives energy.

$\endgroup$
1
$\begingroup$

Radius is usually measured in [m], but for rotational movement it's unit is different to length namely [m/rad]. Hence the unit for torque is [Nm/rad]. Torque times angle will come out as energy. I do not know why radians are omitted, causing confusion for the understanding population.

$\endgroup$
  • 2
    $\begingroup$ Radian by definition, is a dimensionless unit!! It's the length of an arc divided by its radius. $\endgroup$ – Ali Feb 23 '14 at 20:40
  • $\begingroup$ There are other dimensionless units for angles, though, such as Turns -- length of arc divided by the circumference. $\endgroup$ – Gus Apr 3 '15 at 2:40
1
$\begingroup$

$$W = τ\, θ\Rightarrow τ = W/θ$$ So the units of τ must be Joule/Radian. In the SI, since the radian is dimensionless, the units are dimensionally the same, but they are technically different units.

$\endgroup$
0
$\begingroup$

A joule is defined as a specific amount of energy or work done. Torque is neither one of those, so even though the units are the same the meaning of joule cannot be applied in the case of torque.

$\endgroup$
  • $\begingroup$ Sorry, but this doesn't really clarify. $\endgroup$ – SusanW May 19 at 10:49
0
$\begingroup$

It is not uncommon for units of a different physical entity to be used to measure a related physical entity. e.g. distance is generally measured in meters; but it is also measured in light years which is the distance traveled by light in a year. The important thing is that there should be a consistent way to convert one unit to another.

Someone pointed out that Torque is a vector (defined as a cross product) while Work is a scalar (defined as a dot product). However, that can't be "the (only) reason" for different units. Units are defined for "magnitude of a vector", which by itself is a scalar. So, the reason you can't use Joules for torque is because there is no consistent way of converting Newton-meters to Joules and vice versa.

There are 2 types of units viz., the basic/elementary units for mass, distance and time and the compound/derived units such Newton, Joule, etc for physical phenomenon that are derived from the basic units.

So, 1 Newton is the amount of Force required to increase the velocity of 1 Kg of point mass by 1 m/sec in 1 sec, in the direction of the change in velocity. 1 Joule is the amount of work done when a force of 1 Newton moves any point mass by a distance of 1m.

For a unit of Joule to be used for a unit of Torque, you would need a unit of Torque to always perform 1 Joule of work, which is not true.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.