MECHANICS. 251 MECHANICS. known distance AC njiait and O he'mn i|ip inter- section of any axis pt'i|Hndicular to tlitir plane «itli the plane, OC'H.A, hein;,' a line pei|iendieiilar to the forces, the resultant R must have such a position that HW = F,AO + FjCO Snli^tituting for R its value Fj + F,, this be- einMcs or hence ( Fi + Fj) BO = FiAO + FfiO F.AC= (F, + F.)BC F, BC = -AC Fi + F/ and therefore the position of the resultant is given in terms of known quantities. (This ex- presses the obvious fact that the moment of the resultant around an axis through C equals the moment of Fj around the same axis; for the moment of F, around this axis is zero.) In a perfectly similar manner the resultant of two ■ parallel forces in opposite directions may be found. One of the most important illustrations of parallel forces is given by the gravitational ac- tion of the earth on a body. Experiments show that the accelerations of all bodies — all materials and all cpiantities — when falling freely toward the earth at any point on its surface are the same. V/.' Therefore each particle of matter of mass m near the surface of the earth is being acted upon by a force mg, whose direc- tion is toward the centre of the earth. Any large rigid body is, then, under the action of a great number of ]iarallel forces. Their resultant is a vertical force ytf, if M is the total mass of the body. Its centre, i.e. the point tlirough which its line of action always passes, however the body is turned, is called it^ 'centre of gravity' ((j.v. ). Jt may be shown analytically and by experiment that this point coincides with the centre of inertia of the body. This is further evident from the fact that, if a body falls, how- ever it revolves in so doing, its centre of gravity must have the acceleration ri : and this property has been shown to be peculiar to the centre of inertia. It is evident that if a rigid body is under the action of three co-planar parallel forces, one of which is equal and opposite to the resiiltant of the other two, the body is in equilibrium. The conditions then are (II that the algebraic sum of the three forces equals zero; (2) that the algebraic sum of the moments of the three forces Vol. xm.— 17. around any axis equals zero. If any number of eoplanar forces, jmrallel or non-parallel, act on a rigid body their resultant may be found by compoimding them in pairs, as described. If, however, the final pair of forces is a couple, that is, consists of two equal and opposite forces, there is no resultant. The moment of a couple around any axis perpendicular to their plane is the product of either of the forces by their distance apart: this product is called the 'strength' of the couple. The action of a coupie is to make a body rotate about an axis perpen- dicular to its plane and passing through the centre of inertia of the body: and this can be balanced, not by a single force, but by another couple of equal strength, and opposite in direc- tion. A couple is then a rotor. The action on a rigid body of any number of forces in all directions can be reduced in the end to a single force through the centre of inertia and a covtplc: for each force can be re- placed by a parallel force through the centre of inertia and a couple lying in their plane, and so all the forces reduce to the sum of a number of forces all passing through the centre of inertia and to the sum of an equal number of couples each tending to jn'oduce rotation around its own axis passing through the centre of inertia. The dynamics of fluid bodies are considered in Hydhodyx.^mics and PxEUsr.iTics. ouK AXD Energy. Two genera! formuhp were developed in the discussion of translation and rotation, F.r = im.?^ — i- m.%' he= 1 1 <.>■' — iuo^ The first formula may be expressed in words as follows; if a particle 'hose mass is ni is nioving with a speed So in any direction, this will be changed to s in that same direction inider the action of a constant force F in that direction, provided the distance traversed in that time is X as given by the relation Fj; =: 1.4 tils' — %»"«(.". An illustration is afl'orded by an arrow shot from a bow: .s.j = 0. then F.r = X<2ms- Fa; is called the 'work' dtme by the bow, and the qmntity il>»i.s" is called the kinetic energy of IraiiaUition. Any body, not itself in motion, which has the power of producing kinetic energy in another body is said to have potential energy. Thus a bent bow. a compressed spring, a stretched elastic cord, etc., have potential energy. To bend the bow, compress the spring, stretch the cord, etc., a force must be overcome; that is, motion is produced in a direction contrary to the elastic force of the body. The numerical value of the potential energj- is defined as equal to the prod- uct of the force overcome and the distance through which this has been done. i.e. to the 'work done on' the bow, spring, or string. If the spring is compressed by a body falling upon it, the spring gains potential energy since work is done on it and the body loses kinetic energy. (The spring and body together would naturally continue to vibrate up and down, but it may be stipposed here that the spring is caught and held when it is compressed to its greatest extent.) If F is the force of opposition due to the spring: ,T, the distance reqiiired to change the speed of the body of mass m from s to ,■;„; the gain of potential energy of the spring in that distance is Fir, and the loss of kinetic energj- is ipHs- — Im.'o^! where Fr=ims- — msQ^. 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