Page:The New International Encyclopædia 1st ed. v. 13.djvu/275

* MECHANICS. 247 MECHANICS. linear velocity of one body is connected with the presence of the other. In this equation the sum of »i,i,'i and hi,'': and of m,Vi and m.V, is a gcoinrtrical one. for each of the terms is a vector quantity. Owing to the importance of the product mass X linear velocity, it has received a name, linear moinfntiim. See Impact. This law can be expressed in a different way. If the positions of two particles of matter of masses »i, and m. at any instant are given by coordinates x^y^ and v^^.t the coordinates of the 'centre of inertia' (q.v. ) are defined to be DiiXi + m-a^j - 'nijj/i + m-y^ X = ; and !/ ^ ; ■ nil -r "»2 n»i + »i2 Consequently as the particles move, the centre of inertia changes its position. If «i and «. are the com|>oiK'nts along the axis of X of the veloci- ties of the two particles and ic, and ir, their com- ponents along the axis of Y, the components along the axes of X and Y of the velocity of the centre of inertia are »i,i'i + mMi and 1 miU -j- »i..(o. 7)1 1 -1- l)U """ "" nil + TO; But if r, and r, are the actual velocities of the two particles, v, is the geometrical sum of «i and «',. .Consequently, if the actual velocity of the centre of inertia is V, it is the geometrical sum of u and w; that is, " nil -j- m« or (Wi -(- ?)!;) V = nijfi -f «i2i>2 = niiV, -f- m-V,. Since m^ + ni^ does not change in any physical action, v must remain constant in direction and amount, however the velocities of the two par- ticles are altered by their mutual influence. So far as is known this law of inlluence of two bodies can be extended to any number of bodies mutually inliuencing eacli other; that is, if any number of particles of matter of masses, jh,, OTj, etc., are left alone, free from external actions, their velocities, however changed by nuitual re- actions, must satisfy the law that the geometrical sum of the linear momenta remains unchanged, miV, + m.i), + etc. = constant. Expressed in terms of the properties of the centre of inertia of the system of particles, this law is that the centre of inertia of a s3-stem of particles free from external influences moves in a straight line with constant speed. A large solid is of course a special case of a system of par- otides ; and the motion of the centre of inertia of such a body must obey the same laws as does a single particle. This principle of dynamics is kno^ii as the 'conservation of linear momentum.' When this principle is applied to the mutual action of two liodies, it takes the form )H,iii + in^i = constant, where «i, is the mass of one body and i;, is the velocity of its centre of inertia, »»„ is the mass of the second body and t'. the velocity of its cen- tre of inertia, and the summation is a geometrical one. This equation means then, that if »",i', is changed in any w'ay by v^ changing either in direction or in speed, iii;?-, must change at the same time by an amount equal and opposite to that of the change of iii,i. The rates of changes of the two momenta must then be equal and opposite vectors in the same straight line; that is, nua-i = —i,i./u if «, and o™ are the linear .accelerations of the two centres of inertia. (Illus- trations are afforded by a body falling toward the earth, the earth ha.s an acceleration upward ; by a piece of iron attracted to a magnet which is suspended frtie to move, etc.) This may be ex- pressed by saying that under the inlluence of the second body the first has received an acceleration fli. The product »i,f/, is called the 'kinetic reaction' of the body of mass m^ against the given influence, which is equal and opposite the kinetic reaction of the second body against the action of the first. The influence of any body on another of mass m is measured, therefore, by the product of m and the acceleration produced, i.e. ma. If there is a system of many bodies, the action on one due to all the others is the sum of its kinetic reactions against all the actions; that is, it is the product of the mass of that one into the geometrical sum of the accelerations which each in turn of the others by itself would pro- duce — or the actual acceleration of the one. The product of the mass of any body, therefore, by the linear acceleration of its centre of inertia measures the external influences acting on it. These external influences combine to form what is called the 'external force.' In symbols F — ma meaning that if a body of mass m is subjected to a given set of external influences its acceleration is given byF/?n or if bodies of different masses are subject to the same force the accelerations produced vary inversely as the masses. A 'unit force' is such an external action as results in an acceleration of 1 when the mass is I, or an acceleration 2 when the mass is %, etc. If the C. G, S. system is used, the unit of mass is a gram, and a luiit acceleration is a change in one second of the velocity by an amount of one centi- meter jier second ; the unit force on this system is called the 'dyne.' The dTie is so small, being illustrated nearly by the upward force of the hand required to keep a milligram from falling, that a 'megadyne' (or 10° X dyne) is used as a practical unit. There are many kinds of forces (q.v.) : gravi- tation, electrical, magnetic, muscular, elastic, etc. It should not be thought that they are lliiyigs that exist; they are simply numerical values of quan- tities giving the measure of external influences on the motion of a body, e.g. the effect of pulling a string attached to a body, the effect of a mag- net on a piece of iron, etc. Forces are vector quantities and nuiy be compounded or resolved into components. The commonest illustrations of a force are given by a body falling freely to- ward the earth, in which case the acceleration, (j, is a constant for all bodies at any one place on the earth's surface ( see Gravitation ), and so the force on a body of mass m is mr/, and if a body is suspended and kept from falling, there must be an uiiward force inf/ due to the sus- pension ; <7 is nearly 9S0 centimeters per-second per second, or about 32 feet per-second per sec- ond. This product mg is called the 'weight' of the body. One of the most important illustrations of force is sho^vn by uniform motion of a particle in a circle, which may be produced by a string whirling the body in a sling, or by making the body roll around inside a horizontal circular hoop on a smooth table. In the former case the string is said to 'exert a tension' on the particle; ia