321 × 10 × 20 √}{} 6 x 50.3 = 12.1 feet in a second. Also, by Art. 121, the number of revolutions which the paraboloid makes in 1′′ — 12 = 16 ft 2.03, or 121 revolutions per minute. ON PENDULUMS. 122. A Pendulum is either simple or compound. A simple pendulum consists of a particle of matter fastened to the end of a very fine inextensible string, the other end being fastened to a pin, about which it vibrates as a centre of motion. A compound pendulum consists of two or more bodies, or of one body from the figure and extent of which we are not permitted to abstract. 123. The centre of oscillation of a compound pendulum is a point in it at such a distance from the centre of suspension, that a simple pendulum, of a length equal to that distance, will have the same angular velocity with the compound pendulum itself. 124. The centre of percussion, which is generally in the same point as the centre of oscillation, may be explained as follows: In striking any body with a bar or lever, it is always found that if the blow is given at or near the end of the bar, it will jar, or attempt to fly out of the hand; and if the blow is given by that part of the bar near the hand, it will also jar, and attempt to fly from it. Now there evidently must be a point between these two, where, if a stroke is given, the full effect of the blow will be sensible, and the bar will remain at rest, without jarring the hand. This point is called the centre of percussion, or the point in a striking body where, if it strike another, the effect will be most powerful; and as the centre of gravity of a body is a point on which, if suspended, the body would be in equilibrio, so the centre of percussion is a point in which the whole momentum of the moving body is placed to produce the greatest effect. ON THE SIMPLE PENDULUM. 125. It has been found, by many very accurate experiments, that a pendulum which vibrates seconds in the latitude of London is 39 inches in length. This being known, we can find the length of a pendulum which will make any number of vibrations in a given time, as follows: Bring the given time into seconds; then, as the square of the number of vibrations given is to the square of the given number of seconds, so is 393 to the length of the required pendulum in inches. Example. What must be the length of a pendulum, so as to make 80 vibrations in a minute? Here the given time is 60 seconds. Therefore, if the length of a pendulum be required, so as to make a given number of vibrations in a minute, divide 140850 by the square of the number of vibrations given, and the quotient will be the length of the pendulum. Example. What must be the length of a pendulum to make 50 vibrations in a minute? Given the length of a pendulum, to find how many vi brations it will make in a given time. Bring the given time into seconds; then, as the given length of the pendulum is to 39, so is the square of the given time to the square of the number of vibrations, the square root of which is the number sought. Example. If the length of a pendulum be 48 inches, how many vi. brations will it make in a minute? The given time is 60 seconds. the square root of which is 54-17 vibrations in a minute. CENTRE OF OSCILLATION AND PERCUSSION. 126. The distance of the centre of oscillation or percussion of any compound pendulum from its centre of suspension, is equal to the sum of the products of each body into the square of its distance from the centre of suspension, divided by the sum of the products of each body into its distance from that centre. Thus, if any number of bodies, A, B, C, &c. and their respective distances from the centre of suspension, a, b, c, &c. be given, then the distance of the centre of oscillation A a2 + B b2 + C c2 from the centre of suspension is A a + B b + C c Let A4 lbs. and its distance from the centre of suspension 4 inches, B 6 lbs. and its distance 2 inches, and C8 lbs. and its distance from the same point 3 inches; 4 × 42 + 6 × 2o + 8 × 3o 64 +24+72 then, 4x4+6x2+8x3 16 + 12 + 24 = 3'; that is, the centre of oscillation is 3 inches from the centre of suspension. The distance of the centres of oscillation and percussion from the axis of motion is as follows, where the axis of motion is at the vertex and in the plane of the figure: In a right line, small parallelogram, and cylinder, the axis of the figure. In a triangle, the axis. In the parabola, of the axis. If a cylinder, of which the altitude is a, and the radius r, be suspended from its vertex, the distance of the centre 2 a 2 of oscillation from the vertex is + 3 2 a If a cone be suspended from the vertex, the altitude of which is a, and the radius of the base r, the distance of 4 a p2 the centre of oscillation from the vertex is + 5 5 a In a sphere, r = radius, d = distance of the axis of mo tion from its centre; then the distance of the centre of oscillation from the axis of motion is d + 2 r2 5 5 d If the sphere be suspended by a point in its surface, then the distance of the centre of oscillation from that point If a cylinder, of which the radius is 4 inches and altitude 1 foot, be suspended by its vertex; required the length of a simple pendulum which will vibrate in the same time. If a globe, the radius of which is 6 inches, be suspended by a point in its surface, required the length of a simple pendulum which will vibrate in the same time. If the globe be suspended by a string 8 inches long, attached to a point in its surface, then the length of the Sincer 6, we have d = 8+ 6 = 14. = 127. Centripetal force is that force or power which tends constantly to impel bodies towards a fixed point or centre. 128. Centrifugal force is that by which bodies would recede from such a centre, were they not prevented by the centripetal force. 129. These two forces are jointly called Central Forces. 130. When a body describes a circle by means of a force directed to its centre, its actual velocity is every where equal to that which it would acquire in falling by the same uniform force through half the radius. 131. This velocity is the same as that which a second body would acquire by falling through half the radius, whilst the first describes a portion of the circumference equal to the whole radius. 132. In equal circles, the forces are inversely as the squares of the times. 133. When the times are equal, the velocities are as the radii, and the forces are also as the radii. |