velocity due to the velocity of G, and cw its velocity due to the rotation of the beam about G; but, the beam being inelastic, the effect of the impact is to destroy the resolved part of O's velocity at right angles to AB; hence v, must be equal to co. We have, then, from (2), Thus we have determined completely the instantaneous motions of the beam after the impact, and the impulsive reaction of the rod at 0. It may be ascertained that, if the original motion be precisely such as our particular figure represents it, on the consummation of the impact, the beam will detach itself from the obstacle and will then move along freely with the velocities vx, v, w, which we have obtained above. In fact we should find, if we were to assume the beam always to touch the obstacle, that the obstacle would have to exert a continuous attraction instead of a reaction. (6) A uniform horizontal stick, falling to the ground by the action of gravity, strikes at one end against a stone: to compare the blow it receives with what it would have received had both ends struck simultaneously against two stones, the blows being supposed to take place at right angles to the stick. The blow it actually receives is half the blow it would have received, on the latter hypothesis, at each stone. (7) One end of a straight brittle rod is held still in the hand, while the other is tapped against a table till the rod snaps to determine the point of fracture. The fracture will take place at a distance from the fixed end α equal to where a is the length of the rod. (8) A thin uniform brittle rod, capable of turning about one fixed extremity, is struck by a given impulse at a given point: to find the point at which there will be the greatest tendency to snap in two. Let be the length of the rod, a the distance of the point of impact from the free end. 1 3 If a be not greater than, the point where the rod is most likely to snap is at a distance from the fixed end equal to 1 If a be greater than 1, the required point coincides with the point of impact. (9) A perfectly inelastic rod slides in the direction of its length down an inclined plane, and eventually strikes a horizontal plane to find the impulses experienced by the two ends of the rod at the instant of impact. If V be the velocity of the rod the instant before impact, m its mass, and a the inclination of the plane, the blows experienced by the upper and lower ends are respectively equal to (10) A hollow circular cylinder, open at both ends, is moveable about a diameter of one end: to find the distance of the line of the centres of percussion from the fixed diameter. Let a be the radius of the cylinder, and b its length: then the required distance is equal to SECT. 2. Single Body. Smooth Surfaces. Determination of Instantaneous Axes of Rotation, &c. (1) A rigid system at rest is struck by any system of simultaneous blows: to determine the position and velocity of the Spontaneous Axis of Rotation, that is, of a straight line, rigidly connected with the system, which, on the application of the blows, has no motion but in the direction of its length. Let the centre of gravity of the system be taken as the origin of co-ordinates: the system of blows may be reduced to three impulsive pressures X, Y, Z, at the origin, along the axes of x, y, z, respectively, and three impulsive couples the moments of which are L, M, N, in the planes yz, zx, xy, respectively. Let V, V, V, be the components of the absolute velocity of a particle dm, (the co-ordinates of which are x, y, z), just after the impacts, parallel to the axes of co-ordinates; V, V, VI, the components of the velocity of the same particle relatively to the centre of gravity; V, V, V, the components of the velocity of the centre of gravity; w,, w, w,, the angular velocities impressed upon the system about the three axes. Now we have, for the motion about the centre of gravity, the which, by substituting for V, V, V, the values given above, become w1Σ (y2+ z3) dm — wΣxydm — wΣxzdm = L w1Σ (22 + x2) dm — wΣyzdm — wΣyxdm = M w ̧Σ (x2+y3) dm − w ̧Σzxdm — wΣzydm = N (3). To simplify these equations, suppose the axes of co-ordinates to coincide with the principal axes through the centre of gravity, and let A, B, C, represent the principal moments of inertia of the system: then we have Again, for the motion of the centre of gravity we have, if m be the whole mass of the system, and therefore, for the components of the absolute velocity of any particle Sm, we shall have Considering V, V, V, as constant, any two of the equations (6) will represent a straight line: multiplying them in order by L M N Α A'B'C' we get, as a condition to which V, V, V, are subject, L.V M.VN.V. L.X M.Y N.Z + + A B = + + C MA mB mC (7). The direction-cosines of the line are, as appears from its but, if the line be the spontaneous axis, these cosines, as is evident from the definition, must also be proportional to From (8) and (9), V denoting the velocity of the spontaneous |