Infinite number of pairs of Force and distance R from AoR

In summary, the conversation discusses the concept of infinity in relation to a mathematical model of Newtonian mechanics. There is a debate about whether or not there is an infinite number of solutions in physical reality, and it is noted that infinity is a mathematical concept and not a physical one. It is also mentioned that there may be an infinite number of points on a line, and the idea of dividing the line and assigning finite values to each point is questioned. It is suggested that the seeming contradiction could be resolved by considering the seamless and continuous nature of combining tangential force and radius. The idea of an infinite number of positions in space is also discussed, with the conclusion that there is no known perfect mathematical model of physical reality.
  • #1
member 731016
Homework Statement
Pls see below
Relevant Equations
Pls see below
For part (b),
1677040507340.png


The solution is
1677040536739.png


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!
 
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  • #2
Callumnc1 said:
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.
 
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  • #3
haruspex said:
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
 
  • #4
Callumnc1 said:
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.
 
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  • #5
Callumnc1 said:
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.
 
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  • #6
Callumnc1 said:
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.
 
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  • #7
PeroK said:
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.
 
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  • #8
haruspex said:
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.
 
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  • #9
PeroK said:
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.
 
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  • #10
haruspex said:
Right, as I wrote in post #2, infinity is a mathematical concept, not a physical one.
It must be both.
 
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  • #11
PeroK said:
It must be both.
There is no known mathematical model of physical reality which is believed to be perfect.
 
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  • #12
Lnewqban said:
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!
 
  • #13
kuruman said:
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 !
 
  • #14
PeroK said:
If you want to propose that there are only a finite number of solutions, then please list them or describe them.
haruspex said:
Conversely, if you insist on there being an infinite number of positions in space, prove that.
PeroK said:
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 said:
Right, as I wrote in post #2, infinity is a mathematical concept, not a physical one.
PeroK said:
It must be both.
haruspex said:
There is no known mathematical model of physical reality which is believed to be perfect.
Thank you for your replies @PeroK and @haruspex !
 

FAQ: Infinite number of pairs of Force and distance R from AoR

What is the significance of the infinite number of pairs of force and distance from the axis of rotation (AoR)?

The concept of an infinite number of pairs of force and distance from the axis of rotation (AoR) is significant in understanding torque. Torque, or rotational force, is the product of force and the perpendicular distance from the AoR. This principle allows for multiple combinations of force and distance to achieve the same torque, which is crucial in mechanical design and physics.

How does the relationship between force and distance from the AoR affect torque?

Torque is calculated as the product of the force applied and the perpendicular distance from the AoR (Torque = Force x Distance). This means that for a given torque, if the distance from the AoR increases, the required force decreases, and vice versa. This inverse relationship allows for flexibility in applying forces at different points to achieve the same rotational effect.

Can you provide an example of how different pairs of force and distance can produce the same torque?

Consider a torque of 10 Newton-meters (Nm). This torque can be achieved by applying a force of 2 Newtons at a distance of 5 meters from the AoR (2 N x 5 m = 10 Nm). Alternatively, it can be achieved by applying a force of 5 Newtons at a distance of 2 meters from the AoR (5 N x 2 m = 10 Nm). Both pairs result in the same torque of 10 Nm.

Why is it important to understand the infinite number of pairs of force and distance in practical applications?

Understanding the infinite number of pairs of force and distance is crucial in practical applications such as engineering and biomechanics. It allows engineers to design systems that can achieve the desired torque with different configurations of force and distance, optimizing for factors like material strength, space constraints, and mechanical efficiency.

How does the concept of infinite pairs of force and distance relate to leverage and mechanical advantage?

The concept is directly related to leverage and mechanical advantage. By increasing the distance from the AoR, a smaller force can be used to achieve the same torque, effectively providing a mechanical advantage. This principle is used in tools like wrenches and levers, where a longer handle allows for easier application of force to rotate an object.

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