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Suppose we have a rod of length L and suppose we have a motor on one end, suppose the motor exerts a torque on the rod when it spins.

Is this torque independent of the length of the rod?

Basically, I don't understand why a linear force consists of a force directed at the CM and a torque about the CM but which depends on the distance to the CM while torque equation does not appear to be distance dependent.

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suppose the motor exerts a torque on the rod when it spins.

So the rod rotates around its long axis (like a driveshaft)?

Is this torque independent of the length of the rod?

Mostly, yes. The torque will depend on the motor (it may be incapable of delivering torque above a certain value), and the load on the rod (if there is nothing on the other end, it will spin rapidly instead of develop a large torque. It would be similar to trying to push a snowflake with a force of 1N. You can't do that with your arm.

torque equation does not appear to be distance dependent.

If you just deal with torques as torques (as you would with a motor), there is no need to worry about the distance. But if you want to decompose torques into explicit forces, you have to take into account where they act.

But that is the distance from the axis of rotation. You asked about the length of the rod. A rod could be very long, yet all the forces on it are very near the axis of rotation. The length doesn't matter. But if you try to twist a screwdriver while holding the handle (a cm or two away from the axis) or if you twist it from the shaft (a few mm away from the axis), you will find that you are able to develop much more torque due to that extra distance from the axis. It doesn't matter in this case if the screwdriver is 6cm long or 25cm long.

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  • $\begingroup$ What I am getting at is, I don't understand why torques on a rigid body sum when you don't explicitly factor in position. To make it concrete, suppose you are in a plane again with the rod, and you have one motor on each side. When the motors spin they exert a torque on the rod - why can this torque be summed? $\endgroup$ May 16, 2021 at 2:23
  • $\begingroup$ (I don't know that I quite get your question yet. Feel free to edit your initial question to add to it). The effect of torque on a body (like the effect of forces on a body) is linear. Twice the torque, twice the acceleration. As such, we can simply sum the total torque (just like we can sum the total forces when considering linear momentum and acceleration). $\endgroup$
    – BowlOfRed
    May 16, 2021 at 2:27
  • $\begingroup$ @FourierFlux torque is a vector quantity and vectors of the same dimensions can be summed. It doesn’t matter where they are, as long as you don’t change the configuration of the vector you can pretty much take it anywhere for calculation. (Also, the last paragraph of your original question is a little confusing. Would be better if you edit it for more clarity :p) $\endgroup$
    – Natru
    May 16, 2021 at 5:06

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