Page:The American Cyclopædia (1879) Volume XI.djvu/330

318 MECHANICS accelerated rectilinear motion ; 2, the compo- sition and resolution of forces ; 8, centrifugal force ; 4, the pendulum ; 5, the mechanical powers, or the theory of machines. Statical and dynamical principles will be treated in conjunction, as the subjects generally embrace both. I. THE LAWS OF MOTIOX. These laws were pretty well recognized if not established before the Principia of Newton was written. Galileo, Kepler, Descartes, "Wren, Halley, Ilooke, and Huygens had successively advanced toward a comprehension of them, the two works of the last named, on the impact of bodies and on centrifugal force, containing as- sumptions, if not direct statements, of what are known as the three laws of motion. The mission of Newton was more to generalize these laws and apply them to the solution of the motion of the heavenly bodies. His Prin- cipia commences with a statement of the laws of motion, and it is in the form there given that they are generally known. They are as follows: Law 1. Every body continues in a state of rest or of uniform motion in a straight line unless acted on by some exter- nal force. This law results from the prop- erty of inertia, by which matter cannot give itself- or deprive itself of motion. Law 2. Change of motion is proportional to the force impressed, and is in the direction of the line in which that force acts. Law 3. To every action there is always opposed an equal re- action. Force may be denned as any cause by which a body is moved, or held in any position, or has its motion changed. It may be the expansive force of steam and gases, animal power, the attraction of gravitation, or electricity. Inertia is that property of mat- ter which offers resistance to any force tend- ing to change its state of rest or of motion, and it is an element of the greatest impor- tance in mechanics, requiring consideration in every calculation where change of motion takes place. A body occupying a fixed place in space would be in a state of absolute rest ; but ordinarily a body is said to be at rest when it is stationary with regard to surrounding bodies. A body is in absolute motion when moving from one point to another in space, and in relative motion when it is regarded as moving with respect to some other body. The velocity of a moving body is the distance it travels in a tfivcn time, the units of space and time being the foot and the second. Velocity may be uniform or variable. It is uniform when the body moves through equal spaces in equal times. Variable motion may be regular or ir- regular, and it may be accelerated or retarded. When it is accelerated in a constant ratio, It is said to be uniformly accelerated; and when in like manner retarded, it is said to be uni- f irmly retarded. Momentum and Impact. The force of matter in motion is called its mo- mentum, and sometimes quantity of motion, and is equal to the mass or quantity of matter multiplied into its velocity; thus m=q xr, the unit of mass being generally considered the pound avoirdupois. A cannon ball weighing 100 Ibs. and moving with a velocity of 1,200 ft. per second would have a momentum of 120,000 Ibs. or 60 tons. When two bodies have equal momenta, their velocities will be in the inverse ratio to their quantities of matter ; that is, if qxv=q f x t v f j then v : v 1 :: q' : q. A force is impulsive when it acts for a moment only, like the stroke of a hammer. When such a force alone acts against a movable body, it necessarily causes uniform motion, a fact which may be shown experimentally by using At- wood's machine, as will be seen further on. When bodies meet by impact, the motion which results depends upon their degree of elasticity and upon their relative momenta. Bodies are elastic when they have the power of restoring their form after compression or expansion. There are no solids which are per- fectly elastic, although glass and steel are near- ly so within certain limits. Permanent gases are perfectly elastic, having the property of expanding after compression to their original bulk, and of being unchanged in their power of resisting pressure. (See ELASTICITY.) Vapors are also perfectly elastic for all . pressures at which the liquids from which they are derived are above their boiling points. (See BOILING POINT.) Bodies are inelastic when they have no power of restitution after compression. Putty is almost per- fectly inelastic. If _ m' an inelastic body, m, w O fig. 1, is moving in the direction m c, and it encounters another inelastic body, m f , which is at rest, the two bodies will after impact move together with a common veloci- ty equal to half that which the body m had before impact, this having imparted half its momentum to m' ; therefore the momentum of the two bodies after impact must be equal to that of m before impact. If v represents the velocity of the body m before, and tf its velocity after impact, then mv=(m+m')v r , and v f =-?-,. When two inelastic bodies, moving in the same direction with unequal velocities, collide, they will move together with a common velocity, and with a momen- tum equal to the sum of their momenta .pre- vious to impact; or mv + mv'=:(m + m')v t , or // wiv + m't)' T . % ' '* If tWO F IG . 2, moving toward each other, collide, they will come to rest if their momenta are m j^ equal ; but if their O - c -- ^ momenta are une- Fl0t 2 . qual, they will after collision move together with a common ve- locity in the direction of the body having the lesser momentum, and with a momen- tum equal to the difference of momenta be- fore impact; or mv m'v 1 = (m+m')v", and