Page:The New International Encyclopædia 1st ed. v. 13.djvu/278

* MECHANICS. 250 MECHANICS. The angle through which the body turns wliile this eli.iiige is going on is given by the formula of kinematics 2a$ — u- — u^; and as o = L/I, the angle L9 = iIo2 — ilUo' The product L is called 'work' ; and the work is said to be done hy the moment if u is in- creasing, and against the moment if u is de- creasing. %I(J^ is called the 'kinetic energy of rotation' of the body whose moment of inertia about a given axis is I and wliose angular speed is u. JIoTiox IN Gexebai.. If the rigid body is not pivoted around a fixed axis, but is free to move in any direction or manner, it will receive, in general, both linear and angular acceleration under the inlluence of a force, e.g. if a body is thrown in the air. { L'ndcr the action of gravity alone there is, however, only linear acereleration, for reasons to be given inniiediately. ) It has been shown that the linear acceleration of the centre of inertia of a body acted on by any forces is the same as that which a particle having a mass equal to that of the body would have under the action of the same forces. A force in general does not have a line of action passing through the centre of inertia; imagine a plane section of the body through the line of action and the centre of inertia: the force will then in general have a moment about an axis through the centre of inertia perpendicular to this ])Iane. Since the translation of the centre of inertia of the body imdcr the action of the force is quite independent of the rotation, the rotation will be exactly as if the above axis is fixed, i.e. if m is the total mass of the body, I its moment of inertia about this particular axis, F the force, and L its moment about the axis, the linear acceleration F of the centre of inertia will be — and the an- »i gular acceleration -=-. So, if the force has its line of action through the centre of inertia, tbere will be no angular acceleration, e.g. tlic action of gravity. If an impulsive force, whose impulse is K and whose lever arm with reference to an axis through the centre of inertia is 7,;, acts upon the body, the velocity of the centre of inertia in the direction of the force will change accord- ing to the formula v — 1 = K/m, and the angu- lar veloeitv about the axis through tlic centre of inertia will be given by the formula u — Uj = -j-. If the body is originally at rest, its centre of inertia will move instantly in the direction of the force with a velocity K/m, and it will in- KA" stantly rotate with an angular velocity -— . If the line of the force is through the centre nf in crtia A- = 0, and there is no angular motion. This fact fumi.shes an experimental method for the determination of the centre of inertia (q.v.). If the linear velocity of the centre of inertia at any instant is r, and if the angular velocity is (J, the entire kinetic energy is '{;»ir' + %Icj'. where m is the total mass and I is the mo- ment of inertia of the body about the axis of iDtation through the centre of inertia. Composition of Fokces — Statics. If several forces are acting on a rigid body there will be produced as a rule both linear and angular accelerations ; it is a problem then to determine what single force, if any, can produce the same result. If such can be found, it is called the 'resultant.' Since, as stated in kinematics, thi most general motion is a 'screw-motion.' it i-^ impossible in general to have a resultant. If. however, the forces all have their lines of action , in one plane, thej' have a resultant excejit in one case to be noted hereafter. Such forces are called 'coplanar.' It is simplest to distinguisli between two groups of pairs of forces, parallil and non-parallel. Two No.v-Pakallei, Coplanar Forces. Thi lines of action of two such forces meet in a point in their plane. Consider a case in which this ] point is in the rigid body on which the two forces are acting. The ell'ect of a force upon a rigid body is evidently the same wherever its point of application is, provided it is in the line of ! action of the force. Therefore the action of the ,' two forces in this case is as if they were bolli applied at that point of the rigid body where ' their lines of action cross. Their resultant is ; then found by constructing their geometrical .sum at this point; for such a force has obvious- ^ ly a translational ellect equivalent to the sum of the ell'ects of the two forces, and it may be shown by simple geometry that its moment around any axis is equal to the sum of the mo- ments of the two forces around that axis, and so its rotational elTect is the same as the combined effects of the two forces. The line of action of the resultant passes through the point of inter- section of- the two forces, but its point of appli- cation can be anywhere iu this line; conse- quently, it is entirely immaterial whether the point of intersection itself is a point of the body or not. It is evident that if the body is under the action of three forces, one of which is equal and opposite to the resultant of the other two. there is no resulting force or moment: that is, there is neither linear nor angular acceleration. Such a condition is called •equilibrium' (q.v.). The stability, instability, etc., of equilibrium are dis- cussed in the article on Kqiilibrhm. Conversely, if a rigid body is in equilibrium un- der the. action of three non-parallel forces, their lines of action must meet in a point, they must lie in one plane, and one must be equal and op- posite to the geometrical sum of the other two. Two Pahallkl Forces. Two i>arallcl forces form a limiting case of two nou-]iarallel coplanar forws whose point of intersection recedes to an infinite distance. Their geometrical sum then be- comes their aliiclnoic sum; if the two forces are in the same direction, their resultant is a force parallel to them, in the same plane, and numeric- ally equal to the sum of their numerical values; if they are in opposite directions, their resultant is a force parallel to them, in the same plane, and numerically equal to the difference of their numerical values. ( For the time being, the ciise is excluded in which the two parallel forces are equal ami opposite: such a combination is called a 'couple,' q.v.). This resultant must have such a position relative to the two forces that its moment about any axis equals the sum of their moments about the same axis. If the forces are as shown in the figure, F, and Fj being at a