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My book "Fundamentals of Physics" by Halliday, Resnick, Walker says:

The net external torque acting on a system of particles is equal to the time rate of change of the system's total angular momentum

Then it says:

Caution: Torques and the system's angular momentum must be measured relative to the same origin. If the center of mass of system is not accelerating relative to an inertial frame, that origin can be any point. However if it is accelerating then it (center of mass) must be the origin. For example, consider a wheel as the system of particles. If it is rotating about an axis that is fixed relative to the ground, then the origin for applying the formula (net external torque acting on a system =time rate of change of the system's total angular momentum) can be any point that is stationary relative to the ground. However, if it is rotating about an axis that is accelerating (such as when the wheel rolls down a ramp), then the origin can be only its center of mass

My question: Suppose a uniform circular disk is rotating about an axis Q that is x meters away from center of mass. It is also accelerating (linear as well as rotational) just like that wheel rolling down the ramp given in example. Can't we apply this formula about axis Q? According to the text given in book "when the system is accelerating then center of mass must be the origin". Here in my question The center of mass is rotating, so how can we use it as origin?

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  • $\begingroup$ @Charles Francis Thank you so much It is helpful but can you tell me what does Mike Stone mean by 3rd point "the point P is moving parallel to the centre of mass". In rotation of a rigid body all points move parallel to points. Why should we only use "center of mass/point parallel to center of mass" as reference in a moving frame. Can't we take Q(which is x meters away from CoM) as origin when the body is moving about that point Q $\endgroup$
    – Forex007
    Commented Aug 29, 2020 at 8:45
  • $\begingroup$ @Mike thank you it helped but I have some doubt. Can't we take point Q as origin? $\endgroup$
    – Forex007
    Commented Aug 29, 2020 at 8:47

2 Answers 2

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The equation $\vec{\tau}_{ext}=\frac{d\vec{L}}{dt}$ is valid for all frames of reference. Contrary to what is mentioned in the book, taking the centre of mass as the origin is not necessary, however, it is the most convenient.

This is because when we take the frame of reference as COM, even if it is accelerating, the pseudo force passes through the COM, and thus the torque due to pseudo forces is zero. If you take some other origin, the pseudo forces will still pass through the COM, and you will have to consider their torques, and that will complicate the problem.

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The equation $\vec{\tau}_{ext}=\frac{d\vec{L}}{dt}$ is valid in any system of reference. That includes non-inertial frames of reference if we consider the pseudoforces that arise in them. It is unclear to me why the book makes that distinction, which in my opinion is wrong.

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