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It has left me a bit confused because at some places the angular momentum/torques are taken about a point, while in others, it's taken about an axis.

Consider the 2 situations:

Situation-1: In a conical pendulum rotating with constant $\omega$, taking the angular momentum about the point from where it has been attached to the roof, yields that the magnitude of the angular momentum is constant, but direction keeps changing.

Situation-2: Here forces were acting on the endpoints of the rectangle and for it to be in equilibrium, Torque was balanced about the diagonal. (The solution specifically mentioned balancing the torque about the diagonal as axis)[The question was about the suspension of cars when one of the tires is raised a bit higher on the pavement.]

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Well consider the equation of moment of momentum (angular momentum) and moment of force (torque)

$$ \begin{aligned} \vec{L} & = \vec{r} \times \vec{p} \\ \vec{\tau} &= \vec{r} \times \vec{F} \end{aligned} $$

But what is $\vec{r}$? It is the location of the percussion axis or line of action relative to the reference point.

Momentum and forces act along a line in space, that when offset from a reference point, cause the moment of momentum or moment of force. The trick is that the reference point is a point in space, but forces and momentum act anywhere along their lines.

This means that if you take any $\vec{r}$ in the direction of $\vec{p}$ or $\vec{F}$ the result is the same (since the cross product removes all parallel components).

The same applies to velocities which are resolved at a reference point, and they are the moment of rotation with $\vec{v} = \vec{r} \times \vec{\omega}$. The motion is a line in space called the rotation axis.

In Summary

In mechanics, there are three lines in space, one for motion, one for momentum and one for forces, and their moments ($\vec{r} \times \vec{\rm line}$) are resolved about a point.

Some of the common points of reference are

  1. The center of mass.
  2. The origin point.
  3. A kinematic joint location.
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