Suppose we have a simple pendulum as shown in figure
.
In this frame, suppose we fix $\theta$ as positive if rotation is at right of axis of symmetry (as depicted in figure) and negative if rotation is at the left of it.
As we know, "magnitude of the torque vector due to weight" is proportional to the sine of the angle between vector distance of the mass from rotation axis and the vector weight.
My question is: this angle between the vectors should be $\theta$. But then we would have:
- $|T| = |l|\cdot|p|\cdot \sin(\theta)$, if $\theta >0$
- $|T| = |l|\cdot|p|\cdot \sin(\theta)$, if $\theta <0$
Is it correct to pose $|\sin(\theta)|$ if $\theta <0$?
If not so, magnitude couldn't be positive as it must be. I ask this because in definition of magnitude of cross product it never mentions the fact that angle between the two vectors could be negative, but must be always between $0$ and $\pi$, and never tells we should pose the absolute value of sine if required, as in this case we seem legitimate to do.
Consider
and
(Sorry, I just flipped it).
The torque due to weight, about the point of suspension, is given as $$\vec \tau = \vec l\times \vec w$$ where $\vec w$ is the weight.
Now, by the right-hand thumb rule, you’ll notice that the torques in both the cases point in opposite directions wrt each other: into the screen in the first case, out of the plane of the screen in the second.
If you decide to measure angles with the vertical line through the point of suspension as the reference line, then you’ll get a negative sign in the second case, which will just simply refer to the fact that the torque is in the opposite direction to that when all quantities were assumed positive, i.e. in the first case.
In both cases, the magnitude of the torque remains the same, i.e. $$|\vec \tau|=|\vec l||\vec w||sin \theta|.$$ regardless of whether $\theta$ is positive or negative.