Infinite number of pairs of Force and distance R from AoR

[ChiralSuperfields]

Homework Statement

Pls see below

Relevant Equations

Pls see below

For part (b),

The solution is

However, is there really an infinite number of pairs physically speaking? It would be very hard, say, vary the force applied by $0.0000001N$ for example.

Many thanks!

 

[haruspex]

Homework Statement:: Pls see below
Relevant Equations:: Pls see below

For part (b),
View attachment 322694

The solution is
View attachment 322695

However, is there really an infinite number of pairs physically speaking? It would be very hard, say, vary the force applied by $0.0000001N$ for example.

Many thanks!

Infinity is a mathematical concept. It is possible that there is no such thing as infinity in the physical world. Infinitely many pairs of numbers satisfy the equation.

 

[ChiralSuperfields]

Infinity is a mathematical concept. It is possible that there is no such thing as infinity in the physical world. Infinitely many pairs of numbers satisfy the equation.

Ahh ok thank you @haruspex ! I guess this is matter of philosophy

 

[Lnewqban]

However, is there really an infinite number of pairs physically speaking? It would be very hard, say, vary the force applied by $0.0000001N$ for example.

If you draw a F versus R graph, a straight sloped line will form.
That line contains all the possible combinations of F and R that induce the same moment of 25.1 N-m shown in the response.
How many points ((F,R)pairs) can be located alone that line?

Please, see:
https://en.m.wikipedia.org/wiki/Point_(geometry)

https://en.m.wikipedia.org/wiki/Line_(geometry)

I believe that we create that apparent contradiction by dividing our line and assigning a finite value to each point.
In reality, the way in which the tangential force and the radius can be combined to result in a unique value of moment or torque is seamless or continuos.

Following the same reasoning, for a fixed value of radius, the magnitude of tangential force that can be applied (for example, when trying to loosen a rebelious nut with a wrench) can continuosly change from a minimum to a maximum value.

 

[kuruman]

However, is there really an infinite number of pairs physically speaking? It would be very hard, say, vary the force applied by $0.0000001N$ for example.

If you think that's very hard (physically speaking), just imagine how hard it would be to mount a 100-kg disk on a fixed axle of zero diameter as implied by the stated ability to apply a force "at any distance ranging from R = 0 to R = 3.00 m from the axis". It boggles the mind.

 

[PeroK]

Ahh ok thank you @haruspex ! I guess this is matter of philosophy

If you want to propose that there are only a finite number of solutions, then please list them or describe them.

 

[haruspex]

If you want to propose that there are only a finite number of solutions, then please list them or describe them.

Conversely, if you insist on there being an infinite number of positions in space, prove that.

 

[PeroK]

Conversely, if you insist on there being an infinite number of positions in space, prove that.

That's inherent in the mathematical model of Newtonian mechanics.

That said, a case could be made for the finiteness of solutions experimentally, if not theoretically.

 

[haruspex]

That's inherent in the mathematical model of Newtonian mechanics.

Right, as I wrote in post #2, infinity is a mathematical concept, not a physical one.

 

[PeroK]

Right, as I wrote in post #2, infinity is a mathematical concept, not a physical one.

It must be both.

 

[haruspex]

It must be both.

There is no known mathematical model of physical reality which is believed to be perfect.

 

[ChiralSuperfields]

If you draw a F versus R graph, a straight sloped line will form.
That line contains all the possible combinations of F and R that induce the same moment of 25.1 N-m shown in the response.
How many points ((F,R)pairs) can be located alone that line?

Please, see:
https://en.m.wikipedia.org/wiki/Point_(geometry)

https://en.m.wikipedia.org/wiki/Line_(geometry)

I believe that we create that apparent contradiction by dividing our line and assigning a finite value to each point.
In reality, the way in which the tangential force and the radius can be combined to result in a unique value of moment or torque is seamless or continuos.

Following the same reasoning, for a fixed value of radius, the magnitude of tangential force that can be applied (for example, when trying to loosen a rebelious nut with a wrench) can continuosly change from a minimum to a maximum value.

Thank you for your reply @Lnewqban !

There will be an infinite number points ((F,R)pairs) located alone that line.

Many thanks!

 

[ChiralSuperfields]

If you think that's very hard (physically speaking), just imagine how hard it would be to mount a 100-kg disk on a fixed axle of zero diameter as implied by the stated ability to apply a force "at any distance ranging from R = 0 to R = 3.00 m from the axis". It boggles the mind.

Thank you for your reply @kuruman !

 

[ChiralSuperfields]

If you want to propose that there are only a finite number of solutions, then please list them or describe them.

Conversely, if you insist on there being an infinite number of positions in space, prove that.

That's inherent in the mathematical model of Newtonian mechanics.

That said, a case could be made for the finiteness of solutions experimentally, if not theoretically.

Right, as I wrote in post #2, infinity is a mathematical concept, not a physical one.

It must be both.

There is no known mathematical model of physical reality which is believed to be perfect.

Thank you for your replies @PeroK and @haruspex !