APPLIED MECHANICS.] MECHANICS 76i) usually four, although two only are shown. Those bobs form sectors of a ring, and are surrounded by a cylindrical casing F. When the bobs revolve with their proper velocity, they are ad justed so as nearly to touch this casing ; should they exceed that velocity, they fly outwards and touch the casing, and are retarded by the friction. For practical purposes their velocity of rotation about the vertical axis may be considered constant. G, G are hori zontal arms projecting from a socket which is capable of rotation about A, and carrying vertical bevel wheels which rest on E and support C, and transmit motion from C to E. There are usually four of the arms G, G with their wheels, though two only are shown. H is one of those arms which projects, and has a rod attached to Us extremity to aet on the throttle-valve of a steam-engine, the sluice of a water-wheel, or the regulator of the prime mover, of whatever sort it may be. When C rotates with an angular velocity equal and contrary to that of E with its revolving pendulums, the arms G, G remain at rest ; but should C deviate from that velocity, those arms rotate in one direction or the other, as the case may be, with an angular velocity equal to one half of the difference between the angular velocity of C and that of E, and continue in motion until the regulator is adjusted so that the prime mover shall impart to C an angular velocity exactly equal to that of the revolving pendulums. There are various modifications of the differential governor, but they all act on the same principle. Division 3. Working of Machines of Varying Velocity. 124. General Principles. In order that the velocity of every piece of a machine may be uniform, it is necessary that the forces acting on each piece should be always exactly balanced. Also, in order that the forces acting on each piece of a machine may be always exactly balanced, it is necessary that the velocity of that piece should be uniform. An excess of the effort exerted on any piece, above that which is necessary to balance the resistance, is accompanied with accelera tion; a deficiency of the effort, with retardation. When a machine is being started from a state of rest, and brought by degrees up to its proper speed, the effort must be in excess ; (vhen it is being retarded for the purpose of stopping it, the resistance must be in excess. An excess of effort above resistance involves an excess of energy oxerted above work performed ; that excess of energy is employed in producing acceleration. An excess of resistance above effort involves an excess of work performed above energy expended; that excess of work is performed by means of the retardation of the machinery. When a machine undergoes alternate acceleration and retarda tion, so that at certain instants of time, occurring at the end of intervals called periods or cycles, it returns to its original speetl, then in each of those periods or cycles the alternate excesses of energy and of work neutralize each other ; and at the end of each cycle the principle of the equality of energy and work stated in sect. 96, with all its consequences, is verified exactly as in the case of machines of uniform speed. At intermediate instants, however, other principles have also to be taken into account, which are deduced from the second law of motion, sect. 89, as applied by the aid of the principles of sect. 90, to direct deviation, or acceleration and retardation. 125. Energy of Acceleration and Work of Retardation for a Shifting Body. Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time A< let its velocity be increased at a uniform rate of acceleration from v l to tv The rate of acceleration will be du i - v l -:-= constant = ; at AC and (p. 698, 104) to produce this acceleration a uniform effort will be required, expressed by p.. frV" r J (71). (The product of the mass of a body by its velocity is called g its momentum ; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.) To find the energy which has to be exerted to produce the accel eration from 7 j to t 2 , it is to be observed that the distance through which the effort P acts during the acceleration is consequently, the energy of acceleration is being proport : onal to the increase hi the square of the velocity, am independent of the time. In order to produce a retardation from the greater velocity r 2 to
- he less velocity v lt it is necessary to apply to the body a resistance,
connected with the retardation and the time by an equation dentical in every respect with equation 71, except by the sub.stitu- ion of a resistance for an eflort ; and in overcoming that resistance he body performs work to an amount determined by equation 72, tutting Kds for Pds. 126. Energy Stored and Restored ly Deviations of Velocity. Thus i body alternately accelerated and retarded, so as to be brought jack to its original speed, performs work during its retardation ixactly equal in amount to the energy exerted upon it during its .cceleration ; so that that energy may be considered as stored during he acceleration, and restored during the retardation, in a manner .nalogous to the operation of a reciprocating force (sect. 117). Let there be given the mean velocity V = ^(Vj + V]) of a body whose weight is w, and let it be required to determine the fluctuation of velocity v 2 v lt and the extreme velocities v : , v. 2 , which that body nust have, in order alternately to store and restore an amount of energy E. By equation 72 we have Zff which, being divided by gives (73). The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the motion of the body, is f? l !l = ^ , (74). V V 2 w; 127. Actual Energy of a Shifting Body. The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by over coming resistance while being retarded from the same velocity of translation v to a state of rest, is (75). 2*7 This is called the actual energy of the motion of the body, and is half the quantity which in some treatises is called vis viva. The energy stored or restored, as the case may be, by the devia tions of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished. 128. Principle of the Conservation of Energy in Machines. The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in sect. 99 for machines of uniform speed. In any given interval during the working of a machine, the energy exerted added to the energy restored is equal to the energy stored added to the work performed. 129. Actual Energy of Circular Translation Moment of Inertia. _ Let a body of the weights undergo translation in a circular path of the radius p, with the angular velocity of deflexion o, so^that the common linear velocity of all its particles is v = ap. Then the actual energy of that body is By comparing this with the expression for the centrifugal force (wa-lpg), it appears that the actual energy of a revolving body is equal to the potential energy Fp/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body. The product wp-/a, by which the half-square of the angular velo city is multiplied, is called the moment of inertia of the revolving 1 30 Actual Energy and Moment of Inertia of Rotation Radius of Gyration.- -See p. 732, 234-237. 131. Examples of Radii of Gyration. See p. /33, 238. 132 Fly-wheels. K fly-wheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the perioi variations in its angular velocity. The principles according to which variations of angular velocity store and restore energy are the same with those of sect. 126 only substituting moment of inertia for mass, and angular for linear e W be the weight of a fly-wheel, R its radius of gyration a a its maximum, o x its minimum, and A-ifo+Oi) its meai velocity. Let 1_ 03 - BI "S A
XV. - 97