Newton's 2nd Law confusion: mass times acceleration is not a force?

[Peter7799]

Grateful if someone could explain why, if Newton's 2nd law says F=ma, I've read warnings and cautions in several physics books that mass times acceleration is not a force. Is it because the equals sign does not mean equals as in 2+2=4, perhaps?

 

[DrClaude]

I've read warnings and cautions in several physics books that mass times acceleration is not a force.

Can you give a specific quote or reference?

 

[Peter7799]

This from Sears & Zemansky's University Physics, 12th edition. Page 118

 

[Peter7799]

Here's another is of the same type and from the same book

 

[PeroK]

I would take these examples as good, practical advice. The point is that acceleration is the result of a force acting on a massive object in an inertial reference frame. And, if we measure all those quantities in consistent units - SI units, for example - then we have an equality between those two vector quantities.

I don't think you want to get bogged down in the mathematical subtleties of $\vec F = m \vec a$. And how you might arrange that formally in terms of vector spaces.

 

[weirdoguy]

If you think ma is a force, than what force is it? What acts on a body with this force? Why? 2nd law just shows the connection between the geometry of a movement (acceleration) and the forces acting on a body. There are plenty of those type of relationships in physics.

 

[haushofer]

If you think ma is a force, than what force is it?

The resulting force.

I find this text really confusing. But perhaps the idea behind it is that one should better write Newton's second law as $a = \frac{F_{res}}{m}$ to stress that acceleration is the result of force, not the other way around. Or maybe people try to identify $m\cdot a$ as an independent force apart from the forces which add up to $m\cdot a$.

Anyway, I wouldn't phrase it that way.

 

[Lnewqban]

Welcome, Peter!

Newton's 2nd law says that
acceleration = net force / mass

Don’t worry about those sentences, the book is just trying to teach you how to properly do free body diagrams, which should only include forces and moments acting on a mass.
The acceleration, if any, comes after all those forces and moments start acting on a body.

 

[weirdoguy]

Or maybe people try to identify m⋅a as an independent force apart from the forces which add up to m⋅a.

As I understand OP, this is the case OP is talking about, i.e. asks why it's not the case.

 

[jbriggs444]

A force ($\vec{F}$) is a rate of transfer of momentum from one body to another due to an interaction other than an exchange of mass. One usually avoids getting that technical an introductory chapter. Instead, we just talk about how hard one pushes or pulls on something.

By contrast, $m\vec{a}$ is the rate of change of momentum of a single point mass (or of a rigid and non-rotating body). We might sometimes generalize to non-rigid or multi-body systems with $\vec{F} = d\vec{p}/dt$

Newton's second law says that the rate of change of the momentum of a body is equal to the net rate at which you transfer momentum to it. When you say it that way, it is a way of stating that momentum is conserved.

 

[Peter7799]

All,
Many thanks for taking the time to offer your explanations. Putting them all together has enabled to grasp the principal that you've all been driving at.
Kind regards,
Peter

 

[Chestermiller]

The force F in F=ma is the net force acting on a body; ma does not apply to each individual force.

 

[Pikkugnome]

My understanding that the F-side of Newton's second law contain forces stemming from interactions. Things like pushing, friction, gravitational pull and so on. These are also governed by Newton's third law. The ma-side connects interactions to change in momentum of a body. Fictional forces do not obey Newton's third law, so if they are included, they should probably be on the ma-side of the equation. I would not call the resultant force a force either, since it is not based on an interaction. At least some care should be taken when the concept it is introduced first time. After basic understanding of Newton's laws, then maybe the laws can be interpreted more deeply.

 

[Chestermiller]

My understanding that the F-side of Newton's second law contain forces stemming from interactions. Things like pushing, friction, gravitational pull and so on. These are also governed by Newton's third law. The ma-side connects interactions to change in momentum of a body. Fictional forces do not obey Newton's third law, so if they are included, they should probably be on the ma-side of the equation. I would not call the resultant force a force either, since it is not based on an interaction. At least some care should be taken when the concept it is introduced first time. After basic understanding of Newton's laws, then maybe the laws can be interpreted more deeply.

Why don’t friction forces obey Newton’s 3rd law?

 

[PeroK]

My understanding that the F-side of Newton's second law contain forces stemming from interactions. Things like pushing, friction, gravitational pull and so on. These are also governed by Newton's third law. The ma-side connects interactions to change in momentum of a body. Fictional forces do not obey Newton's third law, so if they are included, they should probably be on the ma-side of the equation. I would not call the resultant force a force either, since it is not based on an interaction. At least some care should be taken when the concept it is introduced first time. After basic understanding of Newton's laws, then maybe the laws can be interpreted more deeply.

If you accept that force is a vector quantity, then the sum of two forces is also a force.

 

[jbriggs444]

Fictional forces do not obey Newton's third law

Why don’t friction forces obey Newton’s 3rd law?

@Pikkugnome is talking about fictitious forces.

 

[A.T.]

Fictional forces do not obey Newton's third law, so if they are included, they should probably be on the ma-side of the equation.

I have no idea what you mean here.

1) You can bring any term to either side of the equation.

2) The whole point of introducing fictional forces is to make the Fnet = ma (2nd Law) hold in non-inertial frames, when you include those fictional forces in Fnet, as if they were normal forces.

 

[L Drago]

Grateful if someone could explain why, if Newton's 2nd law says F=ma, I've read warnings and cautions in several physics books that mass times acceleration is not a force. Is it because the equals sign does not mean equals as in 2+2=4, perhaps?

See, newton's second law is approved by evidences. Acceleration is the increase in velocity of the body as a result of application of force. Acceleration is not a force.

a = (v-u) /t
acceleration = (final velocity - initial velocity) /time

If we reorder F = ma, we get

F/m = a

And putting general acceleration formula into Newton second law of motion
F = ma
F = m x (v-u) /t
F = (mv - mu) /t


Hence, proved that acceleration is not a force and F = ma equation is absolutely right. Here is another proof

F = dp/dt is the generalized version of newton's second law

p = m x v
F = dp/dt
F = dmv/dt
F = m(dv/dt)
F = m (v/t) Velocity divided by time is acceleration
F = ma

 

[A.T.]

.... mass times acceleration is not a force....

.... acceleration is not a force ....

Spot the difference.

 

[L Drago]

So let me prove that mass time acceleration is a force.

According to generalized version of newton's second law,

F = dp/dt
We know that
P = m x v
F = dmv/dt
F = m (dv/dt)
F = m ( v/t)
We know that v/t = a
F = ma ( Hence proved)

 

[PeroK]

So let me prove that mass time acceleration is a force.

According to generalized version of newton's second law,

F = dp/dt
We know that
P = m x v
F = dmv/dt
F = m (dv/dt)
F = m ( v/t)
We know that v/t = a
F = ma ( Hence proved)

So, why do you think Sears and Zemansky, in their textbook, caution against treating mass times acceleration as a force? That was the whole point of this thread.

 

[L Drago]

So, why do you think Sears and Zemansky, in their textbook, caution against treating mass times acceleration as a force? That was the whole point of this thread.

But in Newton's book Philosophiae Naturalis Principia Mathematica, the second law must have been described with equational proof as Issac Newton, the greatest physicist ever wrote it and gave three laws of motion.

 

[PeroK]

But in Newton's book Philosophiae Naturalis Principia Mathematica, the second law must have been described with equational proof as Issac Newton, the greatest physicist ever wrote it and gave three laws of motion.

The second law is a law/postulate/axiom on which Newtonian mechanics rests. You can't prove it as such.

In any case, the question is more subtle and was discussed earler in the thread. It might be worthwhile thinking about why the authors wrote what they wrote.

 

[L Drago]

I believe that Newton is right. Please prove that f is not equal to product of m and a. I don't mean to be rude. Just want proof..

The second law is a law/postulate/axiom on which Newtonian mechanics rests. You can't prove it as such.

In any case, the question is more subtle and was discussed earler in the thread. It might be worthwhile thinking about why the authors wrote what they wrote.

 

[PeroK]

I believe that Newton is right. Please prove that f is not equal to product of m and a. I don't mean to be rude. Just want proof..

This is what this thread was discussing. I gave my thoughts in post #5.

 

[A.T.]

I don't mean to be rude.

Then read the question and the already given replies properly, instead spamming the thread, while completely missing the point.

 

[L Drago]

This is what this thread was discussing. I gave my thoughts in post #5.

But kindly give mathematical equational proof to describe what you claim that force is not equal to the product of mass and acceleration.

 

[L Drago]

Then read the question and the already given replies properly, instead spamming the thread, while completely missing the point.

Okay thanks I accept the suggestion

 

[jack action]

Please prove that f is not equal to product of m and a.

The fact that a force is mathematically equal to the product of mass times acceleration - i.e. have the same values - doesn't mean they are the same thing.

Going further with that equation, we get $E = mc^2$. This is called more appropriately the mass-energy equivalence because the equation doesn't mean mass and energy are the same thing. And one could extrapolate that this is true because we based this on $F=m\frac{dp}{dt}$ which is also an equivalence and not an equality. (Outside the mathematical sense, that is.)

 

[L Drago]

The fact that a force is mathematically equal to the product of mass times acceleration - i.e. have the same values - doesn't mean they are the same thing.

Going further with that equation, we get $E = mc^2$. This is called more appropriately the mass-energy equivalence because the equation doesn't mean mass and energy are the same thing. And one could extrapolate that this is true because we based this on $F=m\frac{dp}{dt}$ which is also an equivalence and not an equality. (Outside the mathematical sense, that is.)

I was never saying they were thee same thing we were saying they are equal and e = mc² is Einstein's equation of special relativity and here we are talking about force not energy.