An Introduction to the Use of Generalized Coördinates in Mechanics and Physics

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 edition. Excerpt: ...dT d0 If the axes are chosen so that x0 = 0, y0 = 0, and 60 = 0, we have mx = 0, «/ = P, mk26 = aP. Hence the initial velocities are m v1k The velocities of a point on the axis of X at the distance b from the center of gravity are,-t+-£. The point in question will have no initial velocity if b = It follows that the lamina begins to rotate about an instantaneous center in the perpendicular from the center of gravity a2 + /fc2 to the line of action of the force at a distance from a that line and situated on the same side of the line of the force as the center of gravity. This point is called the center of percussion. (i) A wedge of angle a and mass M, smooth below and perfectly rough above, rests on a horizontal plane. A sphere of radius a and mass To is rotating with angular velocity li about a horizontal axis parallel to the edge of the wedge and is placed gently on the wedge. Find the initial motion (v. Art. 9, Ex. 3). Take as coordinates x, the distance of the edge of the wedge from a fixed axis parallel to it in the horizontal plane; y, the distance of the point of contact of the sphere down the wedge; and 0, the angle through which the sphere has rotated. We have T. =----x2 +-(f-2 xy cos a + t0% both before and after the sphere is set down. After the sphere is set down, y--a0 = 0. Before it is set down, x = y = 0 and $ = O. Since the sphere cannot slip, it exerts an impulsive force P up the wedge, and an equal and opposite force P is exerted on it by the wedge at the instant the two bodies come in contact. dT; We have px =-r = + To) X--my cos a, py =-jr = m (y-x cos a), pe = = Tom Our equations are (Jf + To) x--my cos a = m(Jj--x cos a) = rf(-Xl) = and we have also ad Km:a (2), (3), and (4), a + P..--y--x cos a = From (1)...