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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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  • $\begingroup$ Does this answer your question? Is torque as fundamental a concept as force? $\endgroup$ Commented Dec 5, 2020 at 21:02
  • $\begingroup$ See also physics.stackexchange.com/q/516011/and physics.stackexchange.com/a/518467/392 $\endgroup$ Commented Dec 5, 2020 at 21:02
  • $\begingroup$ Well, sort of, yes. I have seen all of these answers before and upon re-reading them multiple times I do think understand them a little better and see what they are saying. I think, and this could be because I'm thinking about this incorrectly, that (linear) Newtonian mechanics and conservation laws should be all that is require to explain the motion of any sort of system of particles. These answers, however, introduce new, angular conservation laws that don't seem to be related to/derived from Newton's linear ones. I'm generally just having trouble seeing why these laws should apply... (1/2) $\endgroup$ Commented Dec 5, 2020 at 21:33
  • $\begingroup$ if they don't have anything to do with newton's laws. $\endgroup$ Commented Dec 5, 2020 at 21:33
  • $\begingroup$ This should help. The angular momentum of a particle with respect to a point O is defined as $\vec r \times m \vec p$. Everything else falls out using Newton's laws. For example, see Symon, Mechanics for the appropriate derivations. $\endgroup$
    – John Darby
    Commented Dec 5, 2020 at 21:56

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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.

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  • $\begingroup$ Okay, so upon re-reading some of the other answers again, I guess It seems fairly intuitive that the line of action or geometry of a force would be conserved in any coordinate system. I guess I just don't understand why this law should be required. I feel like Newton's three laws should be all that is required to explain any sort of interaction of particles. However, I haven't see any sort of analysis of rotation or torque that only implements newton's 3 laws for linear motion. Maybe I'm just too determined to think about things from a linear perspective, as it did appear from some answers... $\endgroup$ Commented Dec 5, 2020 at 23:32
  • $\begingroup$ that rotation and translation were equally fundamental. $\endgroup$ Commented Dec 5, 2020 at 23:32
  • $\begingroup$ @User4758 - that is a fair point when dealing with point particles. But as soon as you create a clump of particles that move together to form a rigid body, Newton's laws acquire a rotational aspect due to the underlying geometry. It is quite easy to consider a system of particles and derive linear and angular momentum and its derivatives to produce the Newton-Euler equations of motion. $\endgroup$ Commented Dec 6, 2020 at 5:45
  • $\begingroup$ Just as I have done in this post about the equations of motion. $\endgroup$ Commented Dec 6, 2020 at 5:46
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    $\begingroup$ Hmm, yeah, I suppose I didn't really consider the geometrical aspect of things. Which is even demonstrated in the link I found. I just thought that because only point particles were being used, their motion should be able to be modeled by all of the forces affecting them according to newton's laws, but didn't consider that the forces affecting them might not be able to be modeled by newton's laws. Thanks for the help, I think I can finally rest my mind, I tend to have a hard time letting things that I don't understand go. $\endgroup$ Commented Dec 6, 2020 at 6:10
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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.

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  • $\begingroup$ Yes, I'm sorry, I should have clarified that I was thinking about the object as a system of point masses and that you were accelerating the specific point mass linearly forward and that it experienced internal forces from other masses in the object. Regarding how we know, I just feel like there should be some sort of derivation directly form newton's laws that shows that an off-center force would accelerate the mass rotationally as predicted by torque. $\endgroup$ Commented Dec 5, 2020 at 22:48
  • $\begingroup$ The only thing that has bothered me is, why should experimental proof be necessary for this. I feel that newton's laws of linear motion should be able to prove this without need for experimentation. I just haven't seen any sort of proof like this. $\endgroup$ Commented Dec 5, 2020 at 23:38

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