Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/714

(382 MECHANICS Resolu tion of velocity. Change of velo city. Hodo- graph. Accelera tion. Accelera^ tion in uniform circular motion. 32 To resolve a velocity is of course a perfectly in definite problem, unless the number of conditions requisite for definiteness be imposed. For, in general, it may be taken as one side of any complete polygon, whether in one plane or not ; and the other sides, all taken in the opposite order round, represent its components. The only cases which we need consider, in which the con ditions are such as to ensure one definite solution, are (1) when a velocity is to be resolved into components parallel and perpendicular to a given line ; and (2) an extension of the same case to components parallel respectively to three Hues at right angles to one another. In case (I) the given velocity is to be taken as the hypotenuse of a right-angled triangle of which one of the sides is parallel to the given line. In case (2) it is to be taken as the diagonal of a rectangular parallelepiped of which the edges are parallel to the three lines respectively. In either case the magnitude of each component is found by multiplying the amount of the velocity by the cosine of the angle between its direction and that of the component ; and the square of the whole velocity is equal to the sum of the squares of the components. We are now prepared to take up the requisite preliminaries for the application of the Second Law. What, in fact, is " change of velocity " 1 The preceding statements at once enable us to give the answer. For let OA (fig. 5) be the velocity of a point at one 4 instant, OB at a succeeding instant. To convert OA into OB, we must compound with it a velocity represented by AB. AB represents the change. Hence if, during any motion whatever of a point, a line OA be constantly drawn from a fixed point O, so as to repre sent at every instant the mag nitude and direction of the velocity of the moving point, the extremity of OA will describe a curve (plane if the original path be plane, but not otherwise, except in certain special cases) which possesses the following important but obvious properties : (1) the tangent at A is the direction of the change of velocity in the original path ; (2) the rate of motion of A is the rate of change of velocity in the original path. Hence in this auxiliary curve, called the HODOGRAPH (q.v.), the velocity represents, in magnitude and direction, what is called " acceleration " in the original path. And, because the acceleration can thus be repre sented as a velocity, the laws of composition and resolu tion of velocities hold good for accelerations also. 33. Hence, if we desire to know the whole acceleration in any case of motion of a point, we need only -find its components in, and perpendicular to, the tangent to the path. That in the tangent has already been found ; it is v or s as in 29. For that perpendicular to the path we may study the simple case of uniform motion in a circle. 34. [f a point move with uniform speed V in a circle, the hodograph is evidently a circle of radius V, and is de- / v A scribed uniformly in / ^ "/ v the same time as the orbit (see fig. 6), Hence the speeds in the two circles are as their radii. Let R be the radius of the Fig. 6. orbit. Then the magnitude of the acceleration in the orbit (the speed in the hodograph) is found from A:V::V:R; that is, A = V 2 /R. Fig. 5. CI The direction of this acceleration, being that of the tangent to the hodograph, is perpendicular to the corre sponding radius of the hodograph, i.e., to the tangent to the orbit. Hence it is along the radius of the orbit and directed inwards to its centre. 35. In other words, to compels mass to describe an un- The so- natural (because curved) path, it must be acted on by a calle <-l force directed towards the centre of curvature of the path. entl j l - We anticipate so far as to introduce here mass and force, f r ce . although, strictly, we are dealing with kinematics. But the student cannot be too early warned of the dangerous error into which so many have fallen, who have supposed that a mass has a tendency to fly outwards from a centre about which it is revolving, and therefore exerts a "cen trifugal force, which requires to be balanced by a "cen tripetal force." The centripetal force is required if the path is to be curved ; it is required for the purpose of producing the curvature, and not because there is any tend ency to fly out from the centre. 36. Thus, in any motion of a point, the whole accel- Compo eration is the resultant of two parts- the first in the ents direction of motion and of magnitude equal to the rate of increase of speed, the second directed towards the centre of curvature and of magnitude as the curvature and the square of the speed conjointly. The sole effect of the first component is to alter the speed, of the second to alter the direction, of the motion. There is no acceleration per pendicular to the osculating plane, because two successive values of the velocity, and therefore also the corresponding change of velocity, are in that plane. 37. A very convenient expression for acceleration which Angula changes the direction of motion is furnished in terms of velocit , what is called the " angular velocity," i.e., the rate at which direction changes. This also is properly a vector, or directed line, perpendicular to the plane in which the change of direction takes place, and of length proportional to the rate at which the angle assigning the direction changes. 38. In the case of uniform motion in a circle of radius R, with speed V, the time of describing the complete circumference (2?rR) is 2-n-R/ V. Hence the angular velocity is V/R, usually denoted by w. Thus the above expression for the acceleration in a direction perpendicular to the path of a point ( 34) may be written in the form po> 2 , where p is the radius of curvature of the orbit and o> the angular velocity of that radius. The direction of this acceleration, as we have seen, is always towards the centre of curvature. 39. The general difficulty of any question concerning acceleration is usually a purely mathematical one, involving only such physical considerations as are required for the formation of the differential equations, and for the deter mination of the so-called arbitrary constants or arbitrary functions involved in the integrals. We will not now discuss the various forms in which the difficulty may present itself, because in the course of the article many of the more important of these will be fully treated in con nexion with motions actually observed among terrestrial or cosmical bodies. 40. We have sufficiently considered ( 27-29) uniform Uuifor acceleration in the line of motion. Let us now consider acceler uniform acceleration in a fixed direction, whether the tlon motion of the point be in that direction or not. This is j^fij the most general case of the motion of an iinresisted line. projectile, on the supposition that its path is confined to a region throughout which gravity is sensibly constant alik^ in direction and in intensity. Two well-known pro perties of the parabola lead to an immediate solution of our problem.

Let fig. 7 represent a parabola, defined completely by