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Two years back, I asked a similar question, but it was closed as needing details of clarity and then was also deleted. However, every translational quantity must have a rotational equivalent. In translation kinematics, we measure length in meters and area in square meters. Their rotational equivalents are the angle in radians and the solid angle in steradians. In translational kinematics, we measure force in millinewtons. We measure energy in millinewton meters or millijoules. The rotational equivalent of force is torque, which is measured in millijoules per radian.

The translational quantities and the rotational equivalents are given in the table below:

Translational Quantity Translational Unit Rotational Quantity Rotational Unit
length meters angle radian
area radians solid angle steradian
mass grams inertia gram square meters per steradian
velocity meters per second angular velocity radians per second
acceleration meters per square second angular acceleration radians per square second
momentum gram meters per second angular momentum gram square meters per second per radian
force millinewtons torque millijoules per radian

In translational kinematics, one millipascal is defined as one millinewton per square meter, which is equal to force divided by area, and could also be equal to one millijoule per cubic meter equal to energy divided by volume. The rotational equivalent of the millipascal must be measured in millijoules per radian per steradian equal to millijoules per cubic radian, but it can only be calculated as torque divided by solid angle.

What quantity commonly used in physics is measured in "millijoules per cubic radian" (the rotational equivalent of the millipascal)?

UPDATE: I have expanded the body of the question while still keeping the existing answers valid. The question now has more details, so it should be reopened now.

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  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$ May 13, 2023 at 7:10
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    $\begingroup$ In addition, there's already an accepted answer which actually answers an original version of your question: by constantly changing the title you take the answer you lessen relevance of the answer you have already accepted, which is not right by @SenorO who took time to answer your original question. $\endgroup$ May 17, 2023 at 12:35
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    $\begingroup$ I’ve reverted this question to the version which was answered previously. I’ve also paused edits for a while. $\endgroup$
    – rob
    May 17, 2023 at 15:26
  • $\begingroup$ @rob I fixed a typo in the question today. The word "quanity" I corrected to "quantity". I have also expanded the body of the question, but this time it validates the existing answers. $\endgroup$ May 24, 2023 at 15:54
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    $\begingroup$ I don’t get it. One does not change a question to validate an answer. The version of the question for which you accepted the answer is the valid question, not any posterior edit to the question. $\endgroup$ May 24, 2023 at 21:34

2 Answers 2

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Your question is based on a faulty premise - that everything must have a rotational analogue.

Pressure simply does not have a rotational analogue. Density doesn't have a rotational analogue, electric charge doesn't have a rotational analogue, and likewise pressure does not. There is no quantity I can think of commonly used in physics that is measured in "joules per cubic radian", though I'm sure you could contrive one if you wish.

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  • $\begingroup$ "all the link you provided is broken/doesn't exist." I removed that link. See the edit history again. $\endgroup$ May 24, 2022 at 4:42
  • $\begingroup$ you mentioned pressure twice :) $\endgroup$
    – peek-a-boo
    May 24, 2022 at 23:45
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I would like to add that there is no conceivable way to reach "cubic radian", which would be the "rotational analogue of a cubic centi/millimeter, i.e. a fraction 1/10 or 1/100 of a meter raised to the power of 3.

The radian is a unit of plane angles in ordinary 3D geometry, that is its definition is based on the assumption that the angle that it's meant to measure is defined in the 2D plane in turn defined by 2 intersecting (segments of straight) lines. When you "go 3D", and remove the assumption that you measure angles in a plane, you measure them in space, where the unit of solid angle (which can be defined as an angle between planes slicing through a sphere) is the steradian which is not a squared (plane) radian. For 4D, 5D etc there is no particular name for the "hypersolid angle".

So there is no way to measure something in squared radians. Thus also not in cubic radians. You can use steradians, defined in terms of an ordinary sphere in 3D. And the sphere (called by mathematicians ball if "full") is the epitome of rotations in ordinary space.

In case you claim, "yes, but acceleration is measured in terms of squared seconds and jerk in cubic/cubed seconds", I respond, well the power of 2 or 3 is a consequence of measuring time in seconds and then measuring quantities already defined in terms of seconds as "a thing per each second every second". So there is no "squared time" it is a change in speed/velocity (i.e. change in distance per each unit of time) per each unit of time.

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  • $\begingroup$ Not unless you add more dimensions, anyway. $\endgroup$
    – keshlam
    Oct 3, 2022 at 17:11

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