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Deductions from the Laws of Motion

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DEDUCTIONS FROM THE LAWS OF MOTION Impulse and Momentum.-2o. We have now given precise meaning to (22), and we proceed to deduce some consequences which follow from this equation, in virtue of kinematical relations which have been shown to hold between acceleration, velocity and "distance." Let us confine attention, in the first place, to motion in a straight line, which we may take to be parallel to Ox. If P is the force in this direction, M the mass of the body considered, and f. the acceleration produced, we have as in (22) by interactions between the masses which compose it ; this is the principle of linear momentum.

We observe that momentum, like velocity, is a quantity which can be resolved and compounded according to the vector law. If u, v, w are the components of a total velocity q, the total mo mentum of the body is measured by Mq, and Mu is the resolved part of this total momentum in the direction of Ox.

"Centre of Mass" of a System.-23. These results can be ex pressed in another way. According to our definition, the re solved part of the total momentum, in the direction Ox, of a system of masses MA, MB, MC, • • • etc. is given by where (Mu)0 and (Mu) i denote the values of the momentum at the beginning and end, respectively, of the interval T. The product PT , viz., the product Jf the force and the time for which it acts, is termed the "time-effect," or impulse. We may express equation (26) in the statement that change of momentum is equal to the impulse of the applied force. When, on the other hand, P varies during the interval considered, this statement will still hold, provided that the term 'impulse' is applied to the Pdt, i.e., to the sum of the "time effects" of the applied force for all parts of that interval.

21. The significance of these ideas is apparent when we come to consider the behaviour of bodies which interact. According to the third law of motion (§ 18), if a force P is exerted at any instant upon one of two interacting bodies (A), then a force —P is exerted at the same instant upon the other interacting body (B). Let MA and denote the mass and velocity of A: then we

have, as in (25), i.e., the total momentum of two bodies is not affected by inter action.

If external forces act in addition, these will produce effects which are represented by (26). If PA is the external force on A, and the external force on B, we deduce from this equation that i.e., the total impulse of the externally applied forces is equal to the change produced in the total momentum of a system of two masses (A and B). It is not difficult to see that this statement may be generalized for a system containing any number of masses.

22. Next, instead of motion in a straight line, let us consider motion of the most general type. The resultant force P on the body can be resolved, at any instant, into three component forces X, Y, Z, and (§ 16) we may write in which the suffixes o and i relate to the beginning and end, respectively, of the displacement considered.

If X has a constant value throughout the displacement, the integral on the left of (38) is equivalent to X (xi —x0), i.e., to the product of the force and of the distance through which it acts. We call this quantity the space-effect of X, or the work done by X, in the displacement considered : if the displacement had been opposite in sense to the force, so that this product had had a negative value, we should have said that work was done against the force. When X varies during the displacement, we may say that the total work done by X is the sum of its space-effects for all parts of that displacement ; i.e., the total work will still be represented by the integral in (38).

25. If we confine attention, in the first place, to motion in a straight line parallel to Ox, the quantity 2 Mug is one-half the product of the mass and of the square of its resultant velocity. This quantity is termed the kinetic energy of the moving mass. Accordingly, in this case, we may express equation (38) by saying that the work done by the applied force is equal to the increase of kinetic energy.

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