XI. Motion of the Centre of Gravity of two or 125. We have explained in the Statics what is meant by the centre of gravity of a body or a system of bodies; and have shewn that for a given body or system there is only one centre of gravity. If a change takes place in the position of any body of the system, there is a corresponding change in the position of the centre of gravity of the system; and thus we are led to consider the motion of the centre of gravity of two or more bodies. 126. Having given the velocities of two bodies estimated in any direction, required the velocity of their centre of gravity estimated in the same direction. Suppose m and m' the masses of the bodies; let their distances from a fixed plane at a certain instant be a and a' respectively; then the distance of the centre of gravity from the fixed plane is -; see Statics, Arts. 119 and 146. ma + m'a' Let the velocities of the two bodies estimated perpendicular to the plane be b and b'; then at the end of a time t the distances of the bodies from the fixed plane are a+bt and a+b't respectively. Therefore the distance of the centre of gravity from the fixed plane This shews that the distance of the centre of gravity from the fixed plane increases uniformly with the time; and that the velocity of the centre of gravity perpendicular mb + m'b' to the fixed plane is m+m 127. In the preceding Article we have assumed that the two bodies have uniform velocities in the assigned 292 MOTION OF THE CENTRE OF GRAVITY. direction; but the result may be easily extended to the case in which the velocities are not uniform. For the time t may be as short as we please; and if the velocities of the bodies are really variable in the assigned direction, no error will ultimately arise from regarding them as uniform for an indefinitely short time. Thus we have the following general result: the velocity of the centre of gravity of two bodies estimated in any direction at any instant is found by dividing the momentum of the system estimated in that direction at that instant by the sum of the masses. 128. The result just enunciated for the case of two bodies is true for any number of bodies; the mode of demonstration is the same as that given for two bodies. 129. The motion of the centre of gravity of two bodies is not affected by the collision of the bodies. First suppose the collision to be direct. Let m and m' be the masses of the bodies, u and u' their velocities before impact, v and v' their velocities after impact. The velocity of the centre of gravity, by Art. 126, mu+m'u' before impact, and after impact; is m+m' and these are equal by Art. 103. mv + m'v' m+m' Next suppose the collision to be oblique. Let m and m' be the masses of the bodies, u and u' their velocities before impact, a and a' the angles which the directions of motion make with the line of impact; let v and v' be the corresponding velocities, and B and B the corresponding angles after impact. The velocity of the centre of gravity, estimated in the direction of the line of impact, by Art. 126, is before impact, and mu cos a + m'u' cos a' m+m' mv cos B+m'v' cos B m+m after impact; and these are equal by Art. 114. The velocity of the centre of gravity estimated in the direction at right angles to the line of impact, by Art. 126, is mu sin a+ m'u' sin a' before impact, and m + m mv sin ẞ+m'r' sin B' m+m' after impact; and these are equal by Art. 114. Thus the component velocity of the centre of gravity in two directions is the same after impact as before; and therefore the resultant velocity is the same in magnitude and direction after impact as before. 130. It follows from the investigation of Art. 126, that if two bodies move in straight lines, each with uniform velocity, then their centre of gravity moves also in some straight line, with uniform velocity. Hence we may establish the following proposition: the centre of gravity of two projectiles, which are moving simultaneously, describes a parabola. For suppose at any instant that gravity ceased to act; then each body would move in a straight line with uniform velocity, and so would also the centre of gravity. The effect of gravity in a given time is to draw each body down a vertical space which is the same for each body, and which varies as the square of the time; and the centre of gravity is drawn down through the same vertical space. Hence, by reasoning as in Art. 51, we find that the path of the centre of gravity is a parabola. 131. By the method of Arts. 126 and 127, we may establish the following result: If f and f' be the accelerations, estimated in any direction, of two moving bodies, whose masses are m and m' respectively, the acceleration of the centre of gravity of the two bodies estimated in the mf+m'f same direction is m+m' And this result may be extended to the case of any number of bodies: see Art. 128. 1. A body weighing 4 lbs., and another weighing 8 lbs. are moving in the same direction, the former with the velocity of 8 feet per second, and the latter with the velocity of 2 feet per second: determine the velocity of the centre of gravity. 2. Equal bodies start from the same point in directions at right angles to each other, one with the velocity of 4 feet per second, and the other with the velocity of 5 feet per second: determine the velocity of the centre of gravity. 3. In the system of Art. 89 supposing the initial velocity zero, find the velocity of the centre of gravity at the end of a given time. 4. A heavy body hanging vertically draws another along a smooth horizontal plane; supposing the initial velocity zero, find the horizontal and the vertical velocity of the centre of gravity at any instant. 5. Shew that the centre of gravity in the preceding Example describes a straight line with uniform acceleration. 6. In the system of Art. 92 supposing the initial velocity zero, find the velocity of the centre of gravity at the end of a given time resolved parallel to the two planes. 7. Shew that the centre of gravity in the preceding Example describes a straight line with uniform acceleration. 8. Two balls are dropped from two points not in the same vertical line, and strike against a horizontal plane, the elasticity being perfect: shew that the centre of gravity of the balls will never re-ascend to its original height, unless the initial heights of the balls are in the ratio of two square numbers. 9. Three equal particles are projected, each from one angular point of a triangle along the sides taken in order, with velocities proportional to the sides along which they move: shew that the velocity of the centre of gravity estimated parallel to each side is zero; and hence that the centre of gravity remains at rest. 10. P, Q, R are points in the sides BC, CA, AB reBP CQ ᎪᎡ = CP AQ BR spectively of the triangle ABC, such that = shew that the centre of gravity of the triangle PQR coincides with that of the triangle ABC. XII. Laws of Motion. General Remarks. 132. We propose in the present Chapter to make some general remarks concerning the Laws of Motion. It is not necessary that a student should devote much attention to this Chapter on his first reading of the subject. He should notice the points which are here considered, and when in his subsequent course he finds any difficulty as to these points he can examine the remarks which bear upon the difficulty. 133. We will here repeat the Laws of Motion. I. Every body continues in a state of rest or of uniform motion in a straight line, except in so far as it may be compelled to change that state by force acting on it. II. Change of motion is proportional to the acting force, and takes place in the direction of the straight line in which the force acts. III. To every action there is always an equal and contrary reaction: or the mutual actions of any two bodies are always equal and oppositely directed in the same straight line. It is manifest that instead of Laws of Motion it would be more accurate to call these statements, Laws relating to the connexion of force with motion. 134. We have already observed that the motion of a body here considered is of that kind in which all the points of the body describe curves identical in form, though varying in position. For example, when we speak of the motion of a falling body we mean such a motion that every point of the body describes a straight line. The motion which is |