# Why is torque not measured in Joules?

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. Energy is 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.

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Minor note: Torque is usually given by $rF \sin \theta$, not just $rF$, unless the angle is always $90$ degrees of course because $\sin 90 = 1$. –  Joe Sep 20 '12 at 23:45
Torque is a vector; energy is not. They just happen to have the same units. –  Christopher A. Wong Sep 20 '12 at 23:46
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. –  Pink Elephants Sep 20 '12 at 23:48
en.wikipedia.org/wiki/Torque#Units –  sdcvvc Sep 20 '12 at 23:49
More on units of torque: physics.stackexchange.com/q/36079/2451 –  Qmechanic Sep 21 '12 at 9:50

## migrated from math.stackexchange.comSep 21 '12 at 0:06

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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

$$W = F \cdot d$$

where $W$ is the work done, $F$ is the force, $d$ is the displacement, and $\cdot$ indicates the dot product. However, torque on the other hand, is defined as the cross product of $r$ and $F$ where $r$ is the radius and $F$ 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 $Nm$ (Newton meters) to note the distinction between torque and energy.

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

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Thanks so much. This explains it really well. –  General Stubbs Sep 21 '12 at 3:25
No problem. Glad I could help! :) –  Joe Sep 21 '12 at 3:30

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.

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Strictly torque is a pseudo-vector, though we don't generally make that distinction in a introductory class. –  dmckee Sep 21 '12 at 2:59
@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. –  Hal Swyers Sep 21 '12 at 11:49

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

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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.)

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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.

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