26. In the general case of motion in three dimensions, we have, corresponding to (38), the additional equations and is now, by our definition, the kinetic energy of the moving mass.
Equation (39) may be expressed by the same formula as (38), if we make a suitable extension (to cover motion in three dimensions) to our definition of "work." In general the displace ment of a body will not be along the line of action of the force which acts upon it. Suppose then that the displacement is from A to B (fig. 9), and that AB is inclined at an angle 0 to the line of action of the resultant force P; let BN be perpendicular to this line of action. If AB is indefinitely small, we may take P to re main constant, both in magnitude and direction, during the dis According to the vector law, AN is the resolved part of the dis placement AB in the direction of P. We take the product (PAN) to be a measure of the work done by P in the displace ment AB; i.e., we now take as our general ized definition of "work" (cf. § 24) the product of the force and of the resolved part of the displacement in the direction of the force. With this definition, accord ing to (40), the integral on the left of equation (39) measures the total work done in the displacement from (xo, Yo, zo) to (x1, zi), so we may assert, generally that in any displacement of a body, the work done by the applied force is equal to the increase in kinetic energy.
27. Now let us imagine that the force on a mass MA is due to interaction with a second mass MB. Then if X is the component force on MA, there will be, at the same instant, a force —X acting on MB. Let dxA be the displacement of MA in a very small interval of time, and dxB the displacement of MB in the same interval. According to § 24, we shall have dxA = change, during this interval, in the quantity VIA — X dxB = change, during this interval, in the quantity 1/1/B and hence, by addition, X(dxA —dxB) = change, during the interval, 2? (40 in the quantity Suppose, in the first place, that the motion of both masses is confined to the direction Ox. Then if MA and MB move through the same distance (so that dxA=dxB), equation (40 may be expressed in the statement that interaction does not affect the total kinetic energy of the two masses ; and it is evident that this conclusion may be generalized for a system containing any number of masses. The total kinetic energy however will be altered by interaction if the distance between A and B does not remain con stant during the displacement : e.g., if a bullet be fired into a block of wood, the forward pressure on the wood is equal to the back ward pressure on the bullet ; but, since the bullet penetrates the wood, its forward displacement in any interval of time is less than that of the wood ; so less work is done by the forward pres sure than is done against the backward pressure, and the total kinetic energy of the wood and bullet is decreased as a conse quence of the interaction.
We are here thinking of bodies so small that their masses may be imagined as concentrated in points. Such bodies are termed
particles, and bodies of finite size are commonly treated, in dy namics, as made up of large numbers of particles (cf. § 53). We see from the foregoing discussion that special simplification will be possible if we assume that the distance between any two of the particles composing a body is invariable, i.e., that the body is rigid; for then we can say that its kinetic energy, like its momen tum, is unaffected by interaction between the constituent particles. This is the basic assumption of rigid dynamics.
32. An argument similar to that of § 9 shows that the orbit will be confined to a single plane through 0. Let v be the resultant velocity of the particle when its distance from the centre of attraction is r, and let be the distance of 0 from AT (fig. i i ), the instantaneous direction of this velocity. Then, since the force on the particle always acts through 0, we know (§ 29) that the moment of momentum of the particle about 0 will not change ; i.e., since the mass is constant, we may write i.e., (according to the vector law), when the resultant force on the particle acts along the line OA. Hence, if a particle is subjected to forces whose resultant always acts through a fixed point 0, its moment of momentum about 0 will remain unchanged.
3o. In the general case we can show, as in (42), that (xY —yX) = Pp', where P is the resultant force on the particle, and pi is the distance of 0 from the line of action of P. The product Pp' is termed the moment about 0 of the resultant applied force P. Thus we may express (43) by saying that the moment of the applied force about any point is equal to the rate of change of moment of momentum about that point.
Again, if the force P on a body A is due to interaction with a second body B, then by the third law of motion (§ 18) there must act, simultaneously, an equal and opposite force ( —P) on B. As in § 2 1 , we may show that the total moment of momentum of two bodies is not affected by interaction, and we may generalize this result for any system of bodies : the resultant (or total) moment about any point of the external forces which act on a system is equal to the rate of change of the total moment of momentum of that system, about the same point this is the prin ciple of angular momentum.