Force and rate of change of momentum

[PeroK]

Is the formula used to calculate the value of a physical quantity the same thing as the quantity itself?
If so then physical laws may be used in place of definitions.

Although your question is not totally precise, I would say yes. (Although, there definitely seems to be some disagreement on this.)

First, the quantity is what you measure. If, within your theory, you can show that two formulas always produce the same numerical value, then you are dealing with the same physical quantity.

An example would be relativistic momentum:

This could be defined as $\gamma mv$

Or, it could be defined as $m\frac{dx}{d\tau}$.

It would be wrong, in my view, to insist that one of these is "really" relativistic momentum and the other just happens to be always numerically equal to it. The theory of special relativity can be used to show that these two are equivalent. Therefore, they can both validly be called "momentum".

 

[BiGyElLoWhAt]

I would agree with this. And maybe that is how this question of mine started. Sometimes it does work out where the math formula matches 100% with the real world(more like pressure), yet other times it is 100% abstract and removed.

Problem and confusion starts when people DO use the math as physical definitions. People do say, Force IS mass time acceleration. Yes this is true mathematically, for a calculation, but not in real work. Force is that which causes a mass to accelerate.

Being in engineering, I have to apply the physics and math back to the real world, so I have to know how to interpret the information. Most probably see this as being anal and picky but yet it does make a difference to have the correct understandings.

So to me, I agree with Pixel, that force is more less a push or pull, define by N2L and calculated as F=MA. I'll just go with that. Sorry I asked, wish I had never thought of all this. Mind feels like a baked potatoe

No worries. Misconceptions are best to be nipped in the bud. It is somewhat abstract, however.
Newtonian mechanics treats the force as the cause of everything, you can see this in Newton's first law. Everything else is a consequence of the existence of a force(s), so to answer your original question as directly as possible, the force is a cause (an interaction) and the change in momentum (of a system) is the result. So the time rate of change of momentum is equal to the force in magnitude and direction, but one is a cause, and the other an effect.
P.S.
Physicists need to be able to map the math back to the real world as well.

 

[David Lewis]

I think the formula (law that a thing obeys) is not the same thing as the thing itself when you define a physical quantity in the general sense.

In the particular sense, however, the equation serves as your definition.

 

[alkaspeltzar]

No worries. Misconceptions are best to be nipped in the bud. It is somewhat abstract, however.
Newtonian mechanics treats the force as the cause of everything, you can see this in Newton's first law. Everything else is a consequence of the existence of a force(s), so to answer your original question as directly as possible, the force is a cause (an interaction) and the change in momentum (of a system) is the result. So the time rate of change of momentum is equal to the force in magnitude and direction, but one is a cause, and the other an effect.
P.S.
Physicists need to be able to map the math back to the real world as well.

Thank you. You don't know how refreshing that is to hear. I will except it as that, that a force is a push/pull and it causes change in momentum. Like you said, everything has a cause and effect. So because of that relationship, we can mathematically/abstractly equate the numbers of force and rate of change of momentum to one another despite they are different physical quantities.

And the more I think about it, that is true with most formulas and things in general. Liberties are taken to help create math that helps figure out the world but I must not take it so literally either.

 

[BiGyElLoWhAt]

It's worth noting that this particular instance is specific to Newtonian physics (force based), and not say Lagrangian or Hamiltonian mechanics, which treat things differently. Each has it's own "fundamental quantity" that everything else is derived from.

 

[PeroK]

It's worth noting that this particular instance is specific to Newtonian physics (force based), and not say Lagrangian or Hamiltonian mechanics, which treat things differently. Each has it's own "fundamental quantity" that everything else is derived from.

In Lagrangian mechanics that would be "generalised force" = "generalised push and pull"; and most certainly not = rate of change of generalised momentum!

 

[A.T.]

we can mathematically/abstractly equate the numbers of force and rate of change of momentum to one another despite they are different physical quantities.

The whole point of physics is to relate different physical quantities to each other.

 

[BiGyElLoWhAt]

The whole point of physics is to relate different physical quantities to each other.

OP wasn't questioning whether or not $|\sum F| = |\frac{dp}{dt}|$ or $\frac{\sum \vec{F}}{|\sum F|} = \frac{\frac{d\vec{p}}{dt}}{|\frac{dp}{dt}|}$ but rather how to interpret the relationship.

 

[sophiecentaur]

Force is a push or pull, measure in Newtons and it is not a rate.

Why does it have to be just one and not both? Threads like this one take up a lot of time and argufying and people get very agitated. Nothing in Science 'is" anything, on its own. Science is all about patterns of relationships and it's full of 'dualities' because of the multiple ways of describing phenomena. It's another example of big-indians and little-endians. Best to sit on the fence, I think.

 

[pixel]

Is velocity the same as change of displacement per unit time, or are they just numerically equivalent?

That is the definition of velocity, so it is the same. I can define force independently of a changing momentum i.e. by measuring the weight of an object using a scale. That tells me the force due to gravity. If I hold the object in my hand and then let go, F=dp/dt tells me how to relate that independently measured/defined force to the subsequent motion of the object.

 

[David Lewis]

Mass could be defined as force divided by acceleration, yet we don't think of mass as containing a rate. It's not necessary for it to move in order to have mass.

 

[Dale]

I think it's in the way that it's taught in introductory mechanics.

I don't know. I was first taught that velocity was the rate of change of position. Then I was taught that acceleration was the rate of change of velocity. Then I was taught that force is mass times acceleration, so the fact that force is a rate was pretty obvious.

I don't think the problem is with the definition of force or whether a force is a push or pull that causes a change in momentum or whatever. I think that the problem is a misunderstanding of what something being a rate means. For some reason he thought that being a push or pull was incompatible with being a rate, and that having a named unit was also incompatible with being a rate, neither of which are correct.

 

[russ_watters]

$F =$ a push or a pull

I like them all, except the last one. But, perhaps that's a mathematical view.

Er, well, you could have actually given an equation for it instead of belittling it!

 

[houlahound]

Everyone understands push or pull, I think it would be cruel to start with force as a rate of change of momentum for beginners.

 

[russ_watters]

I don't know. I was first taught that velocity was the rate of change of position. Then I was taught that acceleration was the rate of change of velocity. Then I was taught that force is mass times acceleration, so the fact that force is a rate was pretty obvious.

That would be fine if that were the only definition of/measure of/application of force. I'm sure most people can think of a handful of different equations for/types of force. How can we describe the constant tension of a spring as a rate? The static friction holding a block on an incline? The force holding a magnet against a refrigerator?

 

[russ_watters]

Everyone understands push or pull, I think it would be cruel to start with force as a rate of change of momentum for beginners.

Agreed.

 

[David Lewis]

First, the quantity is what you measure.

The only requirement is that the quantity is hypothetically measurable. That's why nothing needs to move when you have a force.

 

[PeroK]

That would be fine if that were the only definition of/measure of/application of force. I'm sure most people can think of a handful of different equations for/types of force. How can we describe the constant tension of a spring as a rate? The static friction holding a block on an incline? The force holding a magnet against a refrigerator?

The OP made several points, which some of us have tried to address:

1) Force is a "push or a pull" and is "not a rate".

2) The units of force are Newtons and do not include time, hence force itself cannot be seen as a rate; only the effect of that force could be a rate.

3) In particular, force cannot be rate of change of momentum. That's just plain wrong.

4) Force can, however, be mass times acceleration.

I would ask you this question:

How can the dimensions of the force holding a magnet against a refrigerator be $MLT^{-2}$? If a force doesn't result in motion, how can it be measured in units of $T^{-2}$? And, where do mass and length come in for that matter?

 

[sophiecentaur]

Everyone understands push or pull, I think it would be cruel to start with force as a rate of change of momentum for beginners.

"Cruel" to a primary school child, yes. Adults on PF try to work beyond that level though and you have to be more sophisticated than "Push or pull" if you want to get anywhere with the subject.

 

[sophiecentaur]

How can the dimensions of the force holding a magnet against a refrigerator be MLT−2MLT−2MLT^{-2}? If a force doesn't result in motion, how can it be measured in units of T−2T−2T^{-2}? And, where do mass and length come in for that matter?

If you use a force meter in both cases, you would get the same answer in Newtons.; the force that stretches a spring by so much will also cause an acceleration and the sums will give you the same result Imagine a car being towed with a rope. The force stretching the rope is 1000N and the force accelerating the car is also 1000N. You could measure that force in two ways. It's the same thing that you're measuring.
Do you also have the same problem with acceleration and gravity? There is a Principle of Equivalence at work with both quantities. I think you may be confusing 'familiarity' with quantities that you feel 'directly' with scientific significance.

 

[PeroK]

If you use a force meter in both cases, you would get the same answer in Newtons.; the force that stretches a spring by so much will also cause an acceleration and the sums will give you the same result Imagine a car being towed with a rope. The force stretching the rope is 1000N and the force accelerating the car is also 1000N. You could measure that force in two ways. It's the same thing that you're measuring.
Do you also have the same problem with acceleration and gravity? There is a Principle of Equivalence at work with both quantities. I think you may be confusing 'familiarity' with quantities that you feel 'directly' with scientific significance.

Please read the posts more carefully. I knew if I tried to summarise the OP's view, someone would assume it was my view!

 

[sophiecentaur]

Please read the posts more carefully. I knew if I tried to summarise the OP's view, someone would assume it was my view!

This is a perennial problem on forums like PF but you really don't need to take offence. I was, as always, commenting on the message and not ad hominem (but I see I used the personal pronoun - "you", when I should have written "one". I can't be expected to read through 91 (!) posts to see who is actually responsible for the ideas I come across.
It's good that we are in agreement about the facts of the matter.

 

[PeroK]

This is a perennial problem on forums like PF but you really don't need to take offence. I was, as always, commenting on the message and not ad hominem (but I see I used the personal pronoun - "you", when I should have written "one". I can't be expected to read through 91 (!) posts to see who is actually responsible for the ideas I come across.
It's good that we are in agreement about the facts of the matter.

Yes, it's been a long hard thread!

 

[Dale]

Everyone understands push or pull, I think it would be cruel to start with force as a rate of change of momentum for beginners.

Sure, I am not suggesting starting there, but as soon as you write Newton's second law it is clear that force is a rate.

 

[Dale]

How can we describe the constant tension of a spring as a rate? The static friction holding a block on an incline? The force holding a magnet against a refrigerator?

In each of those cases there are multiple forces with rates of momentum transfer that sum to zero. It may not be a terribly useful concept in those cases, but it also should not be such a surprise either.

 

[BiGyElLoWhAt]

I honestly think this was a semantics question.
Force describes an interaction.
Momentum (and changes of) are a property of matter.
These two cannot be the same, so what must be the case is a cause and effect relationship. The rate of change of momentum w.r.t. time of an object is equal and magnitude and direction to the net force acting on said object, but they are not the same thing. We can equate them mathematically, but an interaction quantity and a property quantity cannot be physically the same thing.

 

[PeroK]

I honestly think this was a semantics question.
Force describes an interaction.
Momentum (and changes of) are a property of matter.
These two cannot be the same, so what must be the case is a cause and effect relationship. The rate of change of momentum w.r.t. time of an object is equal and magnitude and direction to the net force acting on said object, but they are not the same thing. We can equate them mathematically, but an interaction quantity and a property quantity cannot be physically the same thing.

I believe that what you've described is not semantics, but the difference between Physics and Metaphysics. Physics is, essentially, a science of measurement. Yes, you can theorise and use mathematics, but essentially what something "is" in physics is what you measure. Force, like everything else, is ultimately defined by how you measure it.

Metaphysics, on the other hand, is concerned with the fundamental nature of things, so your argument is essentially that force and rate of change of momentum have different intrinsic natures and are different metaphysically.

One example of where metaphysical thinking caused a problem in physics was the development of relativity and the question of "what is time". The great man cut through this by recognising that time is what a clock measures and time has no intrinsic, metaphysical properties. That insight led to special relativity. Without it, the presumed metaphysical nature of time stood in the way of progress towards SR.

 

[BiGyElLoWhAt]

I don't know if I would say that I'm saying something "is" something, intrinsically, unless you consider saying that "force is an interaction quantity" falls into that category.

The semantics here, I believe, is determining whether these two quantities are related via definition, or if it's a cause and effect relationship. I believe that it's the latter.

To expand on what you were saying, we "measure" momentum by measuring velocity and mass, and then calculate momentum. We can also measure changes in these quantities.
We measure force completely differently. For a spring, we measure k, and we measure x. For gravity, we measure m and h, etc.

IMO, $F=\frac{dp}{dt}$ doesn't say that net force is the time rate of change of momentum, it says that a net force induces a rate of change in momentum, and therefore they are "physically" two different things. Additionally, again, not to beat the dead horse, but force describes an interaction, and momentum and changes in momentum describe matter. The latter argument is, IMO, strong evidence that force "is" not the time derivative of momentum, but that the interaction causes the state of matter to change.

On the other hand:
$E=m_0c^2$
IMO says that mass and energy are the same thing. These are two property quantities, that both describe the same piece of matter. So in essence, mass "is" energy.