# Do angular quantities (such as torque) depend upon the distance from the axis or distance from the origin?

One of my books and some of my friends say that torque and other angular quantities depend upon the perpendicular distance from the axis of rotation. This, in effect, would mean that translating your origin along the axis of rotation shouldn't change the torque or angular momentum of a body. Yet in another book, there is an example of exactly this phenomenon.

The other book and several other resources say that torque and other angular quantities depend upon the position of the object relative to the origin of the coordinate system.

To me, the perpendicular distance from the axis interpretation seems more natural.

Otherwise, imagine you have a door, free to rotate about only its hinges. If I place the origin of my coordinate system along the axis, I will measure some torque for a point on the door. Now, if I set my origin at some other location, according to the second interpretation, I would measure a different torque for the same point. This just seems wrong.

• – Buraian Sep 13 '20 at 16:22

You can measure torque about any point you like, so you can pick an arbitrary origin and calculate torque about that origin. The torque of a single force depends on where you measure it from. So if there is not an obvious axis, like a hinge or a pivot, how do you decide where to measure torque from ?

Well, if the net force on an object is zero then the torque on that object is the same regardless of which point you measure it from. So if the forces on an object are a couple (a pair of equal and opposite forces acting at different points) then it can simplify the calculation of the torque of the couple if you put your origin on the line of action of one of the forces, so that you can ignore that force in the torque calculation.

Torque is not invariant under a change in coordinates, so you will measure different torques in different coordinate systems. Same with angular momentum, such that it is true that in a given system of coordinates torque is equal to the rate of change in angular momentum, even if these values are dependent on the origin you choose.

• Torque is a pseudo vector though – Buraian Sep 13 '20 at 16:22
• @Buraian and how would that change the answer? – Wolphram jonny Sep 13 '20 at 16:38
• pseudo-vectors are not really invariants if you shift the co-ordinate system (afaik) – Buraian Sep 13 '20 at 16:40
• @Buraian vectors neither. For instance, the velocity of an object is not invariant across a new system of coordinates that moves relative to the first. – Wolphram jonny Sep 13 '20 at 16:54