Understanding an Elementary Torque Concept

[GaussianCurve]

Homework Statement

All context is given here. I am trying to understand a few things about what Morin is doing.

1) How is proving that f(x) is a linear function, which I understand the proof of, relevant to the claim that the torques at the left end cancel? Also, the introduction of the function makes little sense to be as well. Clearly in this example $f(a+b) = a+b$ and $f(a) = a$ and $f(b) = b$, so why bother?

2) Why does Morin seemingly assume the answer to the problem at the very beginning of the proof (2.8). Claim 2.1 states $F_3a=F_2(a+b)$ to prove, and then he calls it a "reasonable assumption" in the first line, except it's inside a function, which is the same thing as distance.

Homework Equations

Claim 2.1, (2.8), (2.9), (2.10), and the concluding statement: f(x) = Ax.

The Attempt at a Solution

[/B]
Technically there is already a solution in the book, I am just trying to understand it.

I am brainstorming and thinking that we eventually prove that what f(x) is IS the distance, because f(x) = x (so A=1, but he just says A is irrelevent; why not come out and say it's one?), but in the first line he literally says that we are related the "forces and distances," so that thought doesn't make much sense. Also, yes I do see the footnote (3), but I can argue with that, because they aren't being applied at the same point, so I'm not sure why he's even saying that - so that's about where I am at this point.

4. Conclusion

Is this much confusion bad news for me? How can I understand this better and not have to ask you everything? I'd prefer to not brood over this for >1 hour when most people get this instantly.

 

[haruspex]

Clearly in this example f(a+b)=a+b

Why is that clear?
He only assumes that there exists some function f such that in any set-up like the one given the forces and distances are related by F₃f(a)=F₂f(a+b).
Indeed, it is not possible to prove that f(a+b)=a+b.

he calls it a "reasonable assumption"

He only says it is a reasonable assumption that there exists some function f as defined above (though I'm not sure I agree).
You seem to think he is saying that it is a reasonable assumption that $F_3a=F_2(a+b)$

How is proving that f(x) is a linear function, which I understand the proof of, relevant to the claim that the torques at the left end cancel?

Once he has proved it is linear, i.e. f(x)=Ax for some unknown constant A, he can substitute that in (2.8) to obtain F₃Aa=F₂A(a+b).
Cancelling the A's produces $F_3a=F_2(a+b)$, as was to be proved.

 

[Delta2]

I think your confusion arises because you already know what torque is, you already know that if a body is at perfect rest must have sum of forces and sum of torques both equal to zero.

SUPPOSE now that you have never heard of torque, you have absolutely no idea what it is, you don't know that the static condition is that the sum of torques must be zero. All you know is that there must be some equation involving forces and distances. How that equation would look like? The most reasonable assumption for that equation is equation 2.8 for some function f. One could imagine and write down a lot more complex and worst equations involving distances and forces (for example one equation involving time derivatives of the forces and spatial derivatives of the function f like $\frac{dF}{dt}f(x)+F(t)\frac{df}{dx}+...$ or something involving powers like $F^2f(x)+Ff^2(x)+...$ but equations like these are not what we call "reasonable" assumptions) . So one of the most reasonable assumption is an equation like 2.8.

And it is not so obvious that f(a+b)=a+b or f(a)=a ,f(b)=b, it is because in your mind you already know about torques and you say "ah its obvious that the function f is the distance".

 

[Stephen Tashi]

Homework Statement

All context is given here. I am trying to understand a few things about what Morin is doing.

Why Morin is taking that approach is unclear to me. It is interesting mathematics to show that a few assumptions about a function lead to the conclusion that is it a linear function. Perhaps he intends to use this result later in some other context.

Dimensional analysis would suggest that a simple quantity describing a situation involving force and distance would have units of (force)(distance). Does that lead to assuming a formula of the form (force)(function of distance)? - as opposed to fomulae of the form (function of distance)(force) or (function of distance)(function of force)?

 

[GaussianCurve]

I think your confusion arises because you already know what torque is, you already know that if a body is at perfect rest must have sum of forces and sum of torques both equal to zero.

SUPPOSE now that you have never heard of torque, you have absolutely no idea what it is, you don't know that the static condition is that the sum of torques must be zero. All you know is that there must be some equation involving forces and distances. How that equation would look like? The most reasonable assumption for that equation is equation 2.8 for some function f. One could imagine and write down a lot more complex and worst equations involving distances and forces (for example one equation involving time derivatives of the forces and spatial derivatives of the function f like $\frac{dF}{dt}f(x)+F(t)\frac{df}{dx}+...$ or something involving powers like $F^2f(x)+Ff^2(x)+...$ but equations like these are not what we call "reasonable" assumptions) . So one of the most reasonable assumption is an equation like 2.8.

And it is not so obvious that f(a+b)=a+b or f(a)=a ,f(b)=b, it is because in your mind you already know about torques and you say "ah its obvious that the function f is the distance".

Ok, this response mostly makes sense to me, except that equation 2.8 is reasonable. (2.8) can be rearranged to exactly say that $F_3f(a)-F_2f(a+b)=0$, which implies already that the torque is zero, although we are still trying to investigate what f(x) is it still seems to imply that. Why can't they be unequal since we don't know if it's zero yet?

 

[Delta2]

Ok, this response mostly makes sense to me, except that equation 2.8 is reasonable. (2.8) can be rearranged to exactly say that $F_3f(a)-F_2f(a+b)=0$, which implies already that the torque is zero, although we are still trying to investigate what f(x) is it still seems to imply that. Why can't they be unequal since we don't know if it's zero yet?

Well, they could be unequal, however we chose the equation 2.8 to be homogeneous because that is the most reasonable assumption. I mean, when a body is at rest, the sum of forces is equal to zero, so we can do the assumption that the sum of these quantities $Ff(x)$ would also be zero at perfect rest.

 

[GaussianCurve]

Well, they could be unequal, however we chose the equation 2.8 to be homogeneous because that is the most reasonable assumption. I mean, when a body is at rest, the sum of forces is equal to zero, so we can do the assumption that the sum of these quantities $Ff(x)$ would also be zero at perfect rest.

Okay, but doesn't this assumption already imply the conclusion, which is that at any point the torque is zero? We don't know if $\tau=Fx^2$ or $Fe^x$ but it seems like (2.8) is still trying to say the net torque is zero, because he sets them equal to each other. It seems like the first sentence of claim 2.1 is unproven the beginning (and then proven), but it seems the second and third sentence are initially assumed.

Also, his footnote 3 doesn't make much sense, since the forces are not being applied at the same point.

 

[Delta2]

Okay, but doesn't this assumption already imply the conclusion, which is that at any point the torque is zero? We don't know if $\tau=Fx^2$ or $Fe^x$ but it seems like (2.8) is still trying to say the net torque is zero, because he sets them equal to each other. It seems like the first sentence of claim 2.1 is unproven the beginning (and then proven), but it seems the second and third sentence are initially assumed.

Also, his footnote 3 doesn't make much sense, since the forces are not being applied at the same point.

As I already said, the assumption is such as that the sum of the quantities $Ff(x)$ is zero (this is what makes it reasonable) , yes it is implying that the sum of torques(if we call the quantity $Ff(x)$ as "torque") is zero. It is just that we don't know yet that $f(x)=Ax$. We have to use the fact that the sum of forces is zero as well as to take the sum of the $Ff(x)$ quantities around the other end in order to prove that $f(x+y)=f(x)+f(y)$ which then implies that $f(x)=Ax$ if we assume continuity on f.

And I agree about footnote 3.

 

[GaussianCurve]

As I already said, the assumption is such as that the sum of the quantities $Ff(x)$ is zero (this is what makes it reasonable) , yes it is implying that the sum of torques(if we call the quantity $Ff(x)$ as "torque") is zero. It is just that we don't know yet that $f(x)=Ax$. We have to use the fact that the sum of forces is zero as well as to take the sum of the $Ff(x)$ quantities around the other end in order to prove that $f(x+y)=f(x)+f(y)$ which then implies that $f(x)=Ax$ if we assume continuity on f.

And I agree about footnote 3.

All right, I understand it now. Thank you.