# What is force?

1. Jul 26, 2006

what is force anyway?
and i want the most general defenition.

2. Jul 26, 2006

### actionintegral

This is a better question that you think! According to Bertrand Russell, the statement F=ma amounts to nothing more that a truism or circular definition.

3. Jul 26, 2006

### Hootenanny

Staff Emeritus
The most general definition of force is the rate of change of momentum;

$$\sum\vec{F} = \frac{d\vec{p}}{dt}$$

In classical physics (where the mass of a body in motion is constant) this can be expressed in a more familiar form;

$$\sum\vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt} = m\frac{d\vec{v}}{dt} = m\vec{a}$$

However, there are certain circumstances in classical physics such as rocketry where the above expression fails (also when considering relativistic speeds). The original 'general' expression of Newton's second law, however holds in all cases.

$$\sum\vec{F} = \frac{d\vec{p}}{dt}$$

Last edited: Jul 26, 2006
4. Jul 26, 2006

you just defined the resultant force.
and even if it was called force, sigma f is a sum of forces, so such defenition is circular

5. Jul 26, 2006

### arildno

No, it doesn't. Where did you get that idea from?

6. Jul 26, 2006

### actionintegral

Whenever I am thinking about non-relativistice force, I mentally replace the word "force" with "acceleration". Opposing forces are just accelerations that cancel out.

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7. Jul 26, 2006

### Andrew Mason

I think he meant F = mdv/dt fails if $dm/dt \ne 0$. The correct expression is, of course: $$F = dp/dt = mdv/dt + vdm/dt = ma + vdm/dt$$

AM

Last edited: Jul 26, 2006
8. Jul 26, 2006

### lalbatros

These demands are mutually exclusive.
Generality cannot tell you what a force is.
You better break the circle of circular definitions by recognizing as many practical examples as possible. Look in electromagnetism, fluid dynamics, elasticity, astronomy, ...

Hopefully you will forget about these all-out-of-nothing expectations from physics.

Michel

9. Jul 26, 2006

### pmb_phy

The (total) force on a particle equals the time rate of change of that particle's momentum.
If one assumes that F=dp/dt is a law of physics then then one is using circular logicic. However if one defines force as dp/dt the the circularity disappears. The error in login I mentioned is this - Newton's laws are said to hold only in an inertial frame, while an inertial frame is defined as any frame in which Newton's laws hold.In Eddington's words Every body continues in a state of rest or motion in sofar as it doesn't. A typical method today is to define the inertial frame in a way that has nothing to do with with Newton's first two laws, to define mass by Newton's third law, and to use the second as a definition of force.

For more on the details of this method please see

On force and the inertial frame, Robert W. Breheme, Am. J. Phys., 53(10, October 1985.

That is a well known fact in both classical and relativistic mechanics. One need only turn to the Feynman Lectures to verify that fact. In volume I page 12-2, Feyman wrote
Ouch. In my humble opinion that's a bad habit.

Pete

10. Jul 26, 2006

### actionintegral

Ok - but why? What's the problem in setting m=1 everywhere? Or q=1? Or c=1? I'm not being a smart-aleck, I just don't like lugging around a lot of alphabetic luggage. If something is a constant, I set it =1 whereever possible and move on.

11. Jul 26, 2006

### Rach3

I myself set $$\pi \equiv 1$$. But then I'm braver than most.

12. Jul 26, 2006

### rbj

there isn't. it's all a matter of defining units.

that's sorta what they're trying to do with Planck units.

13. Jul 26, 2006

the equasions described above were all about the sum of forces on a body, not the force on one...
and im pretty much sure that the 2 classical forces, gravity and elecricity must have a common base...

14. Jul 26, 2006

### Pythagorean

Though I've been through the general physics courses, I'm going to try and avoid using a physics definition; it won't be too short, but I'm hoping the concept will be simple.

Force can be seen as change. Without force, there would be no physical events. Let's start with space. If you send something into space, you're putting a force on it by setting it into emotion, but the moment your hand leaves the object, and it's free-floating through space, there are eseentially no forces on it (in reality gravity from all the mass in the universe are pulling on it slightly, but not enough to matter) and it will continue on it's path indefinately, unless another force acts on it.

To take the example further, if all the forces in the universe suddenly stopped, here are some of the differences you'd notice (this is not a scientific explaination, but it should help understand the concept of force):

since there would be no electromagnetic force, big objects flying at each other wouldn't crash, they'd just simply pass through each other. The electromagnetic force is responsible for most (if not all) collisions you see on the macro level in your every day life. This is a result of the electrons from two objects repelling each other so greatly that they smash cars and faces and what not when two such objects meet each other.

Since there would be no gravitational force, at the moment all forces stopped acting everything would simply continue on the path it was headed for just before the forces stopped (and pass through other objects)

Because there's no nuclear forces, the same thing would happen with atoms, they'd simply fall apart, each particle of the atom (this is kind of sloppy, viewing the atoms as particles, but bare with me) would continue on whatever path it was on, there would be no turning and changing direction (that's actually considered acceleration, which is porportional to force) and the particles that make up the atom would simply drift off in their own directions.

And remember, since there's no electromagnetic force, none of these are interacting with each other in anyway. Too make a further ridiculous assumption on this already paradoxal model, I'd assume that statistics would allow all the particles that make up the universe would homogonize (assuming a closed universe). In that sense, forces (in addition to causing physical events) are responsible for a heterogenous planet, in which systems are seperate (or 'closed') entities (like you and me and the earth and a capped jug of wine).

My explaination may not be the ultimate one, but I'm hoping that if enough professonal physicists on this forum pick at it, you'll begin tod evelop a more accurate concept of force.

15. Jul 27, 2006

### Andrew Mason

The essence of f=ma is that for a given force, acceleration varies inversely as the mass. This is not circular. A 1 kg falling brick with a string on a pulley pulls a cart along a horizontal surface. The cart accelerates half as fast if I double the cart's mass; three times as fast if I remove 2/3 of its mass. So for a given size of falling brick, ma = constant. If I change the mass of the falling weight, I change the constant. We call that 'constant' the "force".

AM

16. Jul 27, 2006

### arildno

I am shocked by the ignorance of Hootenanny, pmpphy and Andrew Mason (and possibly, Feynman's).
Learn the difference between MATRERIAL systems and geometrical systems.

A MATERIAL system (which consists of the SAME particles over time) is in the classical sense governed by two main laws:
1. Conservation of MASS.
2. A dynamical law known as Newton's 2.law that, due to 1., has two equivalent forms F=ma and F=dp/dt
(where "a" is the acceleration of the center of mass, m the total mass of the system, and p the total momentum of the system)
A GEOMETRICAL system does not contain the same particles over time, and is not governed by either 1 or either of the two forms of 2.

Don't apply laws on systems they are not valid for! :grumpy:

As for proof of my assertion that rocketry is, indeed, governed by F=ma, it suffices to say that a proper MATERIAL system is the rocket+fuel remaining+fuel ejected.

On this system, there are only internal forces working, hence the C.M of this system has zero acceleration.

Another proper material system is the following:
The rocket fuselage+the fuel that remains up to, and including time T.
That system S has mass m(T) (where T can be regarded as a parameter distinguishing between different material systems).
m(T) is a constant, and all particles contained within S accelerates equally with acceleration a(t) up to time t=T.

Let V be the exhaust velocity relative to S, and consider that at time T there is a particle P attached to S with mass dM.
In the interval T, T+dt, P is separated from S, experiencing momentum change dMV, that is, by Newton's 3.law, applying a force -dM/dtV on S at time T

For the time period t<=T, then, S obeys:
$$F(t)=m(T)a(t), F(T)=-\frac{dM}{dt}V$$
T was chosen arbitrarily.
Furthermore, by comparison of different material systems, we see we can introduce a mass function m(t) so fulfilling $\frac{dm}{dt}=-\frac{dM}{dt}$, we can formulate the proper law of motion for the NON-material system rocket+remaining fuel:
$$\frac{dm}{dt}V=ma, \frac{dm}{dt}<0$$
This, of course, is neither the F=ma or F=dp/dt valid for a MATERIAL system, but that is beside the point, since our system isn't a material system in the first place.

F=ma is seen, however, to be equally valid for any particular MATERIAL system you're looking at.

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17. Jul 27, 2006

### loseyourname

Staff Emeritus
That's a very misleading reading of Russell, actionintegral. You might want to provide a little context for that statement of his. He was not trying to say that physical equations are circular.

18. Jul 27, 2006

### pmb_phy

Its possible to choose constants of nature such that c=1. One chooses a system of basic constants and when that is done then other quantities are then defined through those constants. You can choose the mass of a body to be 1 but if there are more than one particles then you have to allow for another particle having the value of two. Same with charge. If one thinks of force as being defined as F = dp/dt then and always remember that this is a definition rather than an equality then one is less likely to confuse F = ma with F = dp/dp. For example: you made the assertion that Opposing forces are just accelerations that cancel out. That is true if and only if you have the very special situation that the mass of each of the two particles are identical. In general what you said here is not true. If the masses are different then the accelerations will be different too.

Consider also the claims made by alrildno about our so-called ignorance of the subject. He states

Here he is defining a "material system" to be that definition such that his assertions are correct. In my extensive readings on the concepts of force and mass I've yet to see such a term used. He goes on to assume
This he states as a postulate of classical mechanics. However one simply does not need to define such a postulate since it can be reduced to a theorem. That theorem is stated as follows; If the total momentum of a system is conserved in all inertial frames of reference then it follows that the total mass of such a system is conserved. For a derivation of this theorem please see
http://www.geocities.com/physics_world/sr/conservation_of_mass.htm
As you can see from this derivation it is the postulate that momentum is conserved, not that mass is conserved. Conservation of momentum is a theorem which follows from Newton's second law as I recall. Actually mass is defined as the m such that mv is conserved. This is a definition based on observation, i.e. (loosely speaking) mass is defined such that momentum is conserved. Momentum is then defined as p = mv.
If Newton's second law follows from "1" then it is not a law but a theorem. aldrino makes the false statement that (, has two equivalent forms F=ma and F=dp/dt). This is a totally invalid statement in that it does not correspond to what is observed in nature. It totally fails for a relativistic particle moving under a force (and fails for non-constant mass systems). (Notice how he avoids mention of relativistic particles? How convenient for him!)
I have to admit that I have no clue on what he means by this since the term is not defined in classical mechanics that I'm aware of and he does not define it here. He goes on to say
Newton's firtst two laws are valid under all circumstances and his third law fails when one gets into particles moving in fields such as the force between two charged particles. The reason being that the field has momentum.

I believe that aldrino's assertion
shows his misunderstanding of how forces is defined and what F = ma means. F = ma is not a definition. It is an equality under certain conditions. This equality fails under relativistic systems which aldrino fails to address.

Pete

ps - If anyone wishes to read the Am. J. Phys. article I refered to above then I'd be glad to e-mail it to them.

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19. Jul 27, 2006

### pmb_phy

When stated as F = dp/dt it is assumed that one understands "F" as what you refer to as "sum of forces" aka "total force."

The sum that you refer to is as follows: If F_21 is the force on body 2 due to body 1 in the abssence of all other forces and F_31 is the force on body one in the abssence of all other forces then the force (what you refer to as "sum of forces") is the quantity F_1 = F_21 + F_31. It must be understood that F_21 or F_31 refer only to what the force would be on object 1 in the absense of all other forces.

Pete

20. Jul 27, 2006

### Hootenanny

Staff Emeritus
Could somebody point me in the direction of a formal definition of both a material and geometrical system, for I have not come across these terms before.

21. Jul 27, 2006

### pmb_phy

You shouldn't assume that such a definition exists which is universally accepted. Actually I've never seen those terms defined and I've been studying physics for 20 years (one never stops studying physics. One merely stops going to school. ) .

That said, one could take "material system" as one consisting of "material". But that requires that I define "material." Of course one could use the term "material" to mean the same thing as "matter" except that "matter" is not a well-defined quantity. Einstein defined the term to include the EM field whereas many people nowadays understand the term as refering to systems which are entirely composed of particles which have non-zero proper mass.

Note: F=dp/dt is not something I created out of nothing. This is the definition of "force" as given in almost all textbooks on classical mechanics. E.g. Feynman, Marion & Thornton, Corben \$ Stehle, and in Jammer's account of the defintion of force as defined by Newton in Jammer's text "Concepts of Force" (i.e. Newton used the term "force" to mean F = dp/dt). A.P. French's text Newtonian Mechanics (page 166 Eq. 6-1) also explains that F dt = dp is how F is found in Newton's Principia. Nowhere in the Principia does F = ma appear. See also page 315 on French.

The other texts which don't use this definition define "force" as F = -grad U where U is the potential energy of the particle. This definition is used in treatments of analytical mechanics. E.g. Lanczos, Landau & Lif****z). You'll see the definition F = dp/dt in all relativistic treatments of mechanics since in that case

$$F = \frac{d(mv)}{dt} = \frac{d(\gamma m_0 v)}{dt}$$

In such case m will not be a constant in time but will vary as $m = \gamma m_0$. The relation F = ma willl not be valid in this case even when the proper mass remains constant in time.

Pete

Last edited: Jul 27, 2006
22. Jul 27, 2006

### Hootenanny

Staff Emeritus
Thanks for the info
I know, that is why I used it in my previous post as it is universally applicable. I didn't think that the definition using potential would be the 'most general' form so I omitted it, my mistake perhaps.

Last edited: Jul 27, 2006
23. Jul 27, 2006

### pmb_phy

Its of limited use. It requires that there exists a function U such that F = -grad U. This is not always the case. The force of friction does not hace such a function associated with it. Velocity dependant forces also cannot be expressed as -grad U. For example; the magnetic force on a charged particle F = qvxB cannot not be written as -grad U since a particle moving in such a field moves with constant potential energy. The Lorentz force is written as

F = dp/dt = q[E + vxB]

The left equality is Newton's second law while the second equality is the Lorentz force law.

Pete

24. Jul 27, 2006

### pmb_phy

Speaking of units - From The Character of Physical Laws, Richard Feynman

Pete

25. Jul 27, 2006

### rbj

i'd love to have a supposed obscenity embedded into my name so that when i introduce myself, i can emphasize the offending syllable and they can't bleep me.

...

Raquel Welsh: "Would you like to pet my pussy?"

Johnny Carson: "Sure, if you move the cat."

...