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Newton's second law in its most general form, says that the rate of a change of a particleâ€™s momentum * p* is given by the force acting on the particle; i.e.,

Now suppose that an external agent applies a force

(56)

According to Newtonâ€™s third law, the particle must apply an equal and opposite force âˆ’**F**_{a} to the external agent. The momentum **p**_{a} of the external agent therefore changes according to

(57)

Adding together equations (56) and (57) results in the equation

(56)

(57)

(58)

The force applied by the external agent changes the momentum of the particle, but at the same time the momentum of the external agent must also change in such a way that the total momentum of both together is constant, or conserved. This idea may be generalized to give the law of conservation of momentum: in all the interactions between all the bodies in the universe, total momentum is always conserved.

It is useful in this light to examine the behaviour of a complicated system of many parts. The centre of mass of the system may be found using equation (55). Differentiating with respect to time gives

(59)

where * v* =

Suppose now that there is no external agent applying a force to the entire system. Then the only forces acting on the system are those exerted by the parts on one another. These forces may accelerate the individual parts. Differentiating equation (59) with respect to time gives

(59)

(60)

where **F**_{i} is the net force, or the sum of the forces, exerted by all the other parts of the body on the *i*th part. **F**_{i} is defined mathematically by the equation

(61)

where **F**_{ij} represents the force on body *i* due to body *j* (the force on body *i*due to itself, **F**_{ii}, is zero). The motion of the centre of mass is then given by the complicated-looking formula

(62)

This complicated formula may be greatly simplified, however, by noting that Newtonâ€™s third law requires that for every force **F**_{ij} exerted by the *j*th body on the *i*th body, there is an equal and opposite force âˆ’**F**_{ij} exerted by the *i*th body on the *j*th body. In other words, every term in the double sum has an equal and opposite term. The double summation on the right-hand side of equation always adds up to zero. This result is true regardless of the complexity of the system, the nature of the forces acting between the parts, or the motions of the parts. In short, in the absence of external forces acting on the system as a whole, *md** v*/

(61)

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