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So, I understand that, when an object undergoes rotational motion, the individual masses undergo/experience various forces and momentums based on their distances from the point of rotation. As such, a mass that is farther away would, if it were to undergo rotational motion, accelerate/experience a greater force than one that is closer.

I also understand, generally, how torque is derived from angular momentum, though I don't see why the cross products of the individual linear momentums of the masses with their distances from the point of rotation need to be taken as it seems repetitive. And, obviously it would make sense that if you applied the same mathematical operation to all of the different values (force, momentum, mass)then they would match up.

What I'm am confused on is, for an object in free space for example, how do you know that a force applied to it at some point is going to have this exact effect on it?

Like, regarding torque, isn't it only representative of the force that an individual mass would have IF it were to accelerate rotationally. But when you apply the force, you are simply accelerating it linearly, and the mass then experiences various internal forces that affect its trajectory from there.

I'm confused on how you know how all of these forces/internal forces will play out/interact with each other, obviously they will all cancel out, but how do you know that they will cancel out in such a way that it creates a net torque about the center of mass equal to the force times its distance from the center of mass? (without assuming that the force itself creates a torque, since my confusion is how we know the force creates a torque)

I feel like I must be missing something obvious as everyone else I see on similar questions understands this concept fine.

Edit: deciding to accept linked answer as it does technically answer the question I asked. Though I still feel that an explanation in terms of Newton's 3 laws for linear motion should be possible, the answers provided have helped me better intuitively understand why linear forces should also exert torques based on other conservation laws which seem to make sense themselves.

Edit: deleting question as I don't think it adds any relevance and I have posted the link I found to other relevant questions.

Edit: Nvm, I can't delete the question.

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2 Answers2

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The conservation of angular items (things with $\vec{r} \times$ in them) are very much related to the translational laws. Take momentum for example, as linear momentum is conserved not only in magnitude and direction (translational laws), but also in the location of momentum. The line of action of a force, or of momentum (also known as percussion axis) is also conserved, and that is why angular momentum is also conserved.

In this post from equation (3) you see that what is conserved in angular momentum is $\vec{r} \times \vec{p}$. And since $\vec{p}$ is conserved by the translational laws, what (3) says is that $\vec{r}$ (or the location in space) is also conserved.

The reason is the forces, momentum, and rotations all act along infinite lines in space and their geometry is conserved when the "moment of" quantities are conserved.

John Alexiou
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when you apply the force, you are simply accelerating it linearly

Only if that force is applied through the centre-of-mass. Otherwise, If the force is applied at any other point in the object in empty space, it will rotate (about its centre-of-mass).

And that rotation is caused by the torque which is created by the force.

To your questions of how we know that a force causes a torque, the answer is that we can measure it. And after becoming fairly sure after many, many experiments that it is a universal pattern that a force causes a torque if not applied through the centre-of-mass of an object in empty space, then we this also trust this to be the case anywhere else where we haven't yet done the experiment.

Steeven
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