Force and rate of change of momentum

[alkaspeltzar]

well I have talked to other people here at work, other engineers and it is all foreign to us. But maybe it is true and we just don't understand it that way. I guess I will just have to accept it and move on.

At the end of the day I know how to calculate force and know how to use the relationship F=MA and that is all that matters

Thanks

 

[256bits]

well I have talked to other people here at work, other engineers and it is all foreign to us. But maybe it is true and we just don't understand it that way. I guess I will just have to accept it and move on.

At the end of the day I know how to calculate force and know how to use the relationship F=MA and that is all that matters

Thanks

In mechanical statics or dynamics, sure F=ma, is good enough concept to begin with.
But, for example,
Force being the rate of change in momentum is a basic starting point for fluid dynamics, where we have changes in velocity from one point in the flow to the next.

 

[BiGyElLoWhAt]

So I suppose you can start with position, take it's rate of change with respect to time, and get velocity.
$\frac{dS}{dt}= v$
then we can take that and multiply it by mass, and get momentum.
$p = m \frac{dS}{dt}$
So momentum itself is already a rate of change, if for some reason we were to define the quantity $mS$ or mass times position, momentum would be the rate of change of this quantity.
It's not taught this way in lower level physics classes, as it's easier to work with the latter F=ma. However, this is also a differential equation, which it's not taught as in lower level physics classes.
Look at a mass on a spring.
Force of spring = -kx
$\sum F = m\frac{d^2x}{dt^2} = -kx \therefore m\frac{d^2x}{dt^2} + kx = 0$
So I suppose the moral of the story is walk before you run, and this is employed in physics classes.

Edit*
If you were looking at a situation such as a block sliding across a surface that has a couple pivot points like such (see attached), and you wanted to know at what time the object was going to tip, it might be useful to define the quantity $mS$ with S being measured from the pivot. Then you could look at it's rate of change and use that to solve for t.

 

[Dale]

I mean we don't think or work with a force as rate

How can you not? Acceleration is a rate and force is proportional to acceleration, so it must be a rate also.

This is the second time recently that this confusion has come up.

EDIT: actually this is the third thread you have started recently on the same topic. I have merged them, please don't start more threads on the same topic

 

[BiGyElLoWhAt]

How can you not?

I think it's in the way that it's taught in introductory mechanics.
You are taught Newton's laws, and you really focus on N2L. Then you talk about energy, and relate it to F.x, but momentum tends to be taught via conservation and lumped in more so with energy than forces. Just my experience, as I took intro and intermediate mechanics not more than 3 or 4 years ago. The difference is night and day.

 

[BiGyElLoWhAt]

Sorry bear with me but I am just not getting it.

So are you saying that the physical force(what we long ago defined as a push or pull) is exactly the same as rate of change in momentum, aka the rate of momentum transfer?

Don't you have to have a force to have a rate of change in momentum? Part of me thinks if a body has acceleration, then there must be a force. Likewise, if a body has a rate of change in momentum, it must have a force causing it...so aren't they two separate things, just related since one can't exist without the other?

And if they are the same, why don't we use one name...why say force if it is really a rate of momentum transfer or visa versa?

I guess I m looking for a simple explanation, please no math at this point.

I think I see the issue.
Momentum is a property of an object, namely $mv$
Force is an interaction quantity, but the magnitude of the net force is equal to the magnitude of the rate of change of momentum. So, in a sense, they are two different "entities".
Momentum is, as stated above, a defined quantity, and net force = ma is derived from that definition, and the fact that they are equal quantities.
They are distinct in the sense that, just to reiterate, momentum is "had" by and object, in other words, momentum is a quantity that describes an object, whereas Force is a quantity that describes an interaction. When a nonzero net force is applied, you get a change in momentum, and the time rate of change is equal to the net force.

 

[russ_watters]

I'm really not seeing why this is a problem. Force itself is just Newtons (N) and isn't a rate. Momentum change (kg-m/s²) is something you can do with force and is a rate. The units are equal as simple derivations with the definition equations will tell you. I don't see why this should be so difficult to believe/accept.

 

[alkaspeltzar]

I'm really not seeing why this is a problem. Force itself is just Newtons (N) and isn't a rate. Momentum change (kg-m/s²) is something you can do with force and is a rate. The units are equal as simple derivations with the definition equations will tell you. I don't see why this should be so difficult to believe/accept.

That is what I was asking all along.

Force is a push or pull, measure in Newtons and it is not a rate. The change in momentum per time is kg-m/s^2--is rate. Force can create momentum change. Therefore, they are physically two different things, but because one relates directly to the other, we can mathematically write an equation stating the magnitude of one equals the other, just in pure calculation.

 

[alkaspeltzar]

I think I see the issue.
Momentum is a property of an object, namely $mv$
Force is an interaction quantity, but the magnitude of the net force is equal to the magnitude of the rate of change of momentum. So, in a sense, they are two different "entities".
Momentum is, as stated above, a defined quantity, and net force = ma is derived from that definition, and the fact that they are equal quantities.
They are distinct in the sense that, just to reiterate, momentum is "had" by and object, in other words, momentum is a quantity that describes an object, whereas Force is a quantity that describes an interaction. When a nonzero net force is applied, you get a change in momentum, and the time rate of change is equal to the net force.

okay, well that makes sense. So are what you saying is that force and rate of change of momentum are physically different things. But because of the relation one has to another, we can mathematically state they are equal, treat as equal for calculations, yet know if when we work with a force is more less the interaction, that push as I have always been taught.

Sorry I do not have the depth as others in physics. This stuff is hard sometimes to grasp which is why I keep asking

 

[BiGyElLoWhAt]

A force is essentially a push or a pull, but a push or a pull (non-zero) always creates a change in momentum, and the product F*t (for constant force).

So basically:
Force between two objects, acts for a time interval, and the change in momentum is equal to F*t for constant forces.
Interaction quantity for a time interval -> change in property quantity.
Force * time -> change in momentum

 

[PeroK]

That is what I was asking all along.

Force is a push or pull, measure in Newtons and it is not a rate. The change in momentum per time is kg-m/s^2--is rate. Force can create momentum change. Therefore, they are physically two different things, but because one relates directly to the other, we can mathematically write an equation stating the magnitude of one equals the other, just in pure calculation.

It's an interesting argument. But, by that logic, force is not the same as mass x acceleration either. Force can cause a mass to accelerate, but the two are physically different things. That's just pure calculation as well.

And pressure is only calculated to be force per unit area. Or, are those two things physically the same? Is pressure physically the same as force per unit area, or are they only related mathematically?

In general, how do you decide when two things are only mathematically related and when they are the same physical thing?

 

[pixel]

Force is a push or pull. The net force (the net push or pull) is numerically equal to the rate of change of momentum. Maybe it's just philosophy at this point, but to me they're not the same thing, they are just numerically equivalent.

 

[PeroK]

Force is a push or pull. The net force (the net push or pull) is numerically equal to the rate of change of momentum. Maybe it's just philosophy at this point, but to me they're not the same thing, they are just numerically equivalent.

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

 

[ulianjay]

Momentum is a property of a system, force describes an interaction between systems. You can define force as the rate of momentum transfer into a system. This has the same units as rate of change of momentum.

When a force is applied to a system momentum is transferred to the system at a rate (and direction) equal to the force.

Whenever you do a Free Body Diagram or a 'force balance' you are really just applying conservation of momentum.

F = dP/dt is equivalent to:

Rate of momentum transfer in - Rate of momentum transfer out = Rate of change of system's momentum

 

[russ_watters]

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

Bad example. Motion (acceleration) is not the only application for force (F=kx?). So I agree with pixel.

 

[PeroK]

Bad example. Motion (acceleration) is not the only application for force (F=kx?). So I agree with pixel.

That gives us four possibilities, at least, for the definition of a force:

$F = ma$
$F = dp/dt$
$F = -\frac{dV}{dx}$
$F =$ a push or a pull

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

 

[BiGyElLoWhAt]

I think the 4th one is definitely legitimate. All of the mathematical representations are correct, as well. I think OP's question was more conceptual, than anything, however.

 

[David Lewis]

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.

 

[BiGyElLoWhAt]

I would say no to that.

 

[alkaspeltzar]

I would say no to that.

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

 

[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.