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Newton's third law of motion says that the internal forces in a system cancel out each other --- what is the system here? what can be considered as a system? and why do the forces cancel out each other's effects?

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what is the system here? what can be considered as a system?

A system can be anything.

For example, if we consider the Earth and Moon as one system, then their mutual gravitational attractions are internal forces, which cancel themselves out. If we considered only the Moon as the system, then the gravitational force from the Earth is not internal anymore.

and why do the forces cancel out each other's effects?

Because, by Newton's 3rd law, whenever there is a force present then there is also always a reaction force of the same size but opposite. Naturally, these cancel each other out if they are both acting on a system.

Those two forces do not act on the same bodies, though. If you kick a bowling ball, then the ball feels your kicking force, while your foot feels the ball's reactions force. But if you consider both ball-and-you as one system, then both forces are included in the system. Each object feels a non-zero net force, making them accelerate (the ball starts moving, while your foot is slowed down). But the system as a whole doesn't feel any net force. It doesn't accelerate. The two parts that the system consists of (ball and you) are accelerating, but the centre-of-mass of this ball-and-you system doesn't accelerate at all.

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There are many general misconceptions about Newton's 3rd law.Read the statement of 3rd law in simple terms:-

"If a particle A exerts a force F on a particle B,then particle B would exert a force --F on A".

We don't know why particle A would exert a force and in counter attack particle B would exert a equal and opposite force.It is a law and we have to follow it just as we follow definitions in mathematics.

So,this law never talks about systems,it talks about particles. You can take anything as a system and a force which is applied by particle of the system on other particle is internal (for multi particle system). It is a consequence of 3rd law of motion that internal forces cancel out in the system. A force applied by an agent other than the system is external force for the system.I am saying again that internal forces cancel each other in a multi particle system as a consequence of 3rd law and we have to follow it. A research is being done in LHC for the proof of statement of 3rd law on gravitons.

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Firstly Newtons 3rd law doesn't mention systems it just says that is object A exerts a force of object B then object B will exert an equal force but in oposite direction on object A.

This is practical physics and as you mention systems lets use the system that is the human body.

Instead of object A use the palm of your hand. For object B use your forehead.

If you place the palm of your hand (object A) on your forehead (object B) you will ferl your forehead exert preasure on the palm of your hand that is equal to the preasure that the palm of your hand is exerting on your forehead. This is a practical example of Newtons 3rd law within the system of a human being.

However due to the Law of Energy Conservation which states energy can neither be created or destroyed the energy used by the forces relating to Newtons 3rd Law of Motion in the Human System are transformed into sound energy which is of course vibrations on a quantum level.

The theorem for this energy conversion is as follows

IF force energy > doh THEN quantum vibration will be (argh)

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Consider a system of two cars colliding. The system consists of two cars. That's what Newton means.

Internal forces are any forces due to bodies inside the system, hence the name. Although for each car, there is a net force due to the other car (as they just collided with each other and there is a contact force), for the system as a whole, nothing really acts on it.

If this is the case, momentum within that system is conserved.

Now if there is an external force (a force by anything that is outside the system acting on bodies in it), momentum conservation law still does not fail (as Bill pointed out in the comments) as the final momentum is simply adding over the impulse of an external force to the initial momentum:

$$p_{final} = p_{initial}+ \int Fdt$$

This is in fact the general expression for momentum conservation in a system. In cases where there is no external force, $F=0$ and we get our familiar $p_{initial}=p_{final}$ relation.

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    $\begingroup$ @AaronStevens ya, I have edited them. $\endgroup$ – Karthik Jan 6 '19 at 16:57
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    $\begingroup$ No, the momentum conservation law is $\vec{p}_{final}=\vec{p}_{initial}+\int \vec{F} \mathrm{d}t$. If $\vec{F}$ is zero, the momentum is constant, but momentum is still conserved even if $\vec{F}$ is not zero. Conservation includes the possible transfer of momentum into or out of a system. Conservation is not the same as constancy. Constancy is a subset of conservation. Momentum conservation law never fails. $\endgroup$ – Bill N Nov 6 '20 at 19:09

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