THEOREM III.—The acceleration of any point P of a body in motion with respect to a moving system is the resultant of (1) the acceleration of P regarded as fixed to the moving system, (2) the acceleration of P relative to the moving system, (3) the Coriolis acceleration 2sva. where Vr is the relative velocity of P and (4 the absolute velocity of the moving system; 2afoo is directed toward the centre of curva ture of the relative path of P unless wo and (Jr are opposite. For other demonstrations consult Routh,
Fig. 2 illustrates this case; u is the given velocity, and the direction but not the mag nitude of v is known; the construction deter mines v completely. The drawing of the two parallels u may be avoided by noticing that u and v have equal projections on the line be tween the two points; this means that the line is inextensible.
II. Two rods AP and BP are hinged at P. the velocities u and se of A and B being given and v, that of P, is required. At P draw the vectors u and v; from the head of u draw a line a perpendicular to AP, and from the head of instantaneous radii, which fact furnishes an obvious method for getting w.
When C lies off the drawing paper proceed thus. Since the body is rigid and w must have equal projections on PR, and v and w equal projections on QR. Draw the projections at R and erect normals at their ends; the head of w will lie at the intersection of these nor mals. This method fails when P, Q, R are col linear. If the instantaneous centre cannot be used note that (a) the projections of u, v, as on PR must be equal, and (b) as the body c3nqot v a perpendicular b to BP. Now P is on the rod AP, therefore if the arrow representing its velocity starts at P it must, by I, end on a; v must also end on b. Hence v goes from P to
the intersection of a and b.
III. Given the velocities of two points of a rigid body to find that of any other point.
In Fig. 4 u and v are given and as is re quired. From equation (1) if the velocity y) of P in Fig. 1 is zero tan e; therefore since i/e is the cotangent of the angle the velocity of 0 makes with X, that point whose velocity is zero at a certain in stant lies on a normal to the velocity of 0. There is evidently only one such point for if there were two the whole body would be in stantaneously at rest. This point is called the instantaneous centre of rotation; it was discov ered by John Bernoulli, (De Centro Spontaneo Rotationis' (1742) but Descartes had previously noticed it in studying the cycloid. In Fig. 4 the normals to and v intersect at C which, being the instantaneous centre, lies on a normal to w. As the body rotates instantaneously about C, the velocities are proportional to the bend, the components of u, v, as normal to PR, at P, Q, R must end on a straight Line; w is then found from its two components.
IV. To find the velocity of any point on a rod moving in a sleeve. There are several cases which are shown in Figs. 5-7. In Fig. 5 the sleeve is a tube free to rotate about a fixed axis; the rod slides in the sleeve and turns with it. Suppose u to he known. The only point, besides P, whose direction of motion is known is that point R on the rod which is also at the axis of rotation of the sleeve: it moves along the rod but has no transverse component. Hence the velocity of R is found as in II, and that of any other point as in III.
In Fig. 6 the sleeve slides on A and turns about the end of B. The centre of the sleeve has two motions: s along and u normal to A, their resultant v being perpendicular to B, so that if one of these three velocities is known the vector triangle in Fig. 6 gives the other two.