Page:The New International Encyclopædia 1st ed. v. 13.djvu/272

* MECHANICS. 244 MECHANICS. name •moment of inertia' was given Zinr' by Eulcr. Xewton gave the principles of mechanics their final form, and since his day there have been no important additions to them. We owe to Newton ( lti4i-1727) the recognition of other forces than weiglit, the general idea of force, and in particu- lar the conception of inertia or mass as a prop- erty of matter distinct from its weight, the gen- eral statement of the principle of the composition and resolution of forces, and the law of action and reaction being equal but opposite. Newton adopted as the proper measure of a force the ac- celeration produced in a given portion of matter; or, in other words, the velocity produced in a given time. According to Iluygens the measure of the force is the square of the velocity produced in a given distance. Among the philosophers who came after Newton and Huygens tlu>re wjis a scliool. following Descartes, who measured forces by the change in m r : another, following Leibnitz, who measured it by the cliange in titv". Thus, to a certain extent one school succeeded Newton; the other. Huygens. The two were shown by D'Alcmbert to be identical, although there was a great controversy for many years concerning their relative merits. KINEMATICS. All ])Ossible motions of any geometrical figure may be divided into two classes, trunslation and rotation. In the former all lines in the moving figure remain parallel to themselves, i.e. the motions of all the points are identical ; in the latter all the points of the figure are describing circles whose centres lie on a straight line called the 'axis.' In the general case the motion of a figure is a combination of translation and rota- tion. Translation. In motion of translation it is necessary to consider the motion of one point of the figure only, as that is the same for all the points. If the figure is moved from one position to another, this disptdcemeiit may be represented by a straight line joining the initial and final positions of any one point of the figure. This line indicates by its direction and its length the dis|)hicement of the whole figure: it is called a vector, and displacement is said to be a vector quantitii because it rc(]uires for its complete un- derstanding a direction and a numerical (piiuitity only, and so can be pictured by a straight line having the proper direction and a length equal to or proportional to the numerical quantity. If the motion of the figure is imiform — that is, if it passes over equal distances in equal in- tervals of time — the rate of motion, or the dis- tance traversed divided by the time taken, is called the linear speed. If the motion is not ini- forni. the linear speed at any iu'ilant is the dis- tance which the figure would move in the next second if its motion were to continue for that interval of time at exactly the same rate as it is at that instant: in mathematical symbols, if A.r is the length of the extremely short distance traversed in the extremely short interval of time A/ immediatelv following the given instant, the St linear speed at that instant is the value of -— in the limit as A' is taken smaller and smaller. Speed is therefore a number. If the speed in a particular direction is considered — that is. if a distinction is made between the motions of fig- ures with the same speed but in different direc- tions — the linear speed in a given direction is called the linear velocity in that direction. Linear velocity is evidently a vector quantity; the linear speed giving the numerical quantity, i.e. the length of the vector. If a figure is given simultaneously two dis- placements, the resulting displacement is evi- dently found by 'adding geometrically' the two components. Thus if Al.i and BC represent the two component displacements, the actual one will be AC. formed liy jilacing BC so as to con- tinue the motion indicated by AB and completing the triangle. (A man walking across the deck of a moving ship illustrates this 'composition' of displacements.) Similarly, if AB and BC repre- sent the linear ve- locities of the two component motions, the actual velocity is represented in di- rection and speed by AC. In a perfectly s i m i 1 a r manner, three, four, etc., vec- tor qualities may be added geometrically. Further, conver.sely, any displacement or velocit.v may be re- garded as made up of two disphieenients or two velocities, the condition being that the two vectors represent- ing the component quantities should form a broken line joining the en<ls of the vector rep- resenting the actual quantity. This is called 'resolution' of displacement or velocity. In re- solving vectors it is nearly always best to take the components so that they are at right angles to each other, for then they are independent of each other, thus if .B is a displacement— or any vector — its 'component in the direction' AF is "the vector AC obtained by dropping a per- pendicular from B ujion .F. AB is equivalent to AC and CB, but CB has no connection with the direction AF, and AC is then that component of AB which indicates how nnieh AB is con- cerned with the direction AF. In mathematical language the component in the direction AF of a vector AB is AB cos (CAB). In general the velocity of a moving figure will not be constant; and the rate of change of the linear velocity at any instant — that is. if ir- is the extremely small change of the velocity in the extremely small interval of time At, the limitinc value of — — is called the linear ac- '^ At rileration at that instant. It ia evident that acceleration being the change in velocity, and therefore the <litrerence between two lines, is it- self a vector quantity: it has a numerical value and a definite direction., and as with displace- ments and velocities, accelerations can be com-