Torque/moment of force fundamentally

Question

I've very much gotten used to calculating the torque about some axis of an object by multiplying the force and the distance from the axis at which the force acts on.

We also know that if two opposite, equal moments (equal torque) act about the same axis, the total moment is zero and therefore the angular acceleration of the object is zero. This means that if we apply a clockwise force at some distance from an axis, and a counter-clockwise force with half the magnitude but twice the distance, the total moment is zero and the object is not accelerating angularly. That is, if the object was initially at rest it stays so with these moments applied.

But why is this true? What is the physical reason for this? Obviously we can define moments mathematically however we choose, but how does nature know that even though the counter-clockwise force is only a half of the clockwise force, it is twice as far from the axis and therefore the object should stay at rest? Is this something that we have just observed to be true about nature or is there a reasoning for this from other physical principles/laws? This has never really been explained to me and I never have been able find a good explanation for it. Obviously we can measure torques and conclude that this principle seems hold always but why do we believe it to be true in general?

I can believe there could be a simple reasoning behind this but I would appreciate if somebody could clarify this for me.

Answer 1

Let's assume the object actually would start to accelerate angularly. This means the torque must have done some amount of work, since the kinetic energy of the object would change. Calculating the work yields: $$\Delta E_{kinetic}=W=\textrm{force}\cdot\textrm{distance}=F_1\cdot r_1\cdot \theta+F_2\cdot r_2\cdot \theta,$$ where $F_2$ is the force at some distance $r_2$ from the axis and $F_1=-2F_2$ is the force twice as big as $F_1,$ but in the opposite direction and at a distance $r_1=\frac{1}{2}r_2$. $\theta$ is the angle by which the object has rotated measured in radians. Substituting this in the work equation above gives: $$W=\theta\left(-2F_2\cdot \frac{1}{2}r_2+F_2\cdot r_2\right)=0$$ This means that the change in kinetic energy is zero, and thus the rotational speed must be unchanged. Said in another way, there is no angular acceleration.