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in a small circular arc, is equal to PN where l is the radius
149. Corol. 1. Hence all the vibrations of a pendulum in a cycloid, whether great or small, are performed in the same time, which time is to the time of falling through the axis, or half the length of the pendulum, as 3.1416 to 1, the ratio of the circumference to its diameter; and hence that time is easily found thus. Put p = 3.1416, and I the length of the pendulum, also g the space fallen by a heavy body in 1" . of time.
1 then wg: NT::1":N the time of falling through !,
1 theref. 1:p::
which therefore is the time
2g of one vibration of the pendulum.
150. And if the pendulum vibrate in a small arc of a circle; because that small arc nearly coincides with the small cycloidal arc at the vertex B; therefore the time of vibration in the small arc of a circle, is nearly equal to the time of vibration in the cycloidal arc; consequently the time of vibration
of the circle.
151. So that, if one of these, g or 1, be found by experiment, this theorem will give the other. Thus, if g, or the space fallen through by a heavy body in 1" of time, be found, then this theorem will give the length of the second pendulum. Or, if the length of the second pendulum be observed by experiment, which is the easier way, this theorem will give g the descent of gravity in 1". Now, in the latitude of London, the length of a pendulum which vibrates seconds, has been found to be 39 inches; and this being written for l in the theorem, it gives PV
=l": hence is
2g found g=1p1=p2 x 39 = 193.07 inches = 16' feet, for the descent of gravity in 1"; which it has also been found to be, very nearly, by many accurate experiments.
152. Hence is found the length of a penduluin that shall make any number of vibrations in a given time. Or, the number of vibrations that shall be made by a pendulum of
given length. Thus, suppose it were required to find the length of a half-seconds pendulum, or a quarter-seconds pendulum; that is, a pendulum to vibrate twice in a second, or 4 times in a second. Then, since'the tinse of vibration is as the square root of the length,
394 1:4:: 391: = 94 inches nearly, the length
4 of the half-seconds pendulum. Again 1:16 :: 395:27 in. ches, the length of the quarter-seconds pendulum.
Again, if it were required to find how many vibrations a pendulum of 80 inches long will make in a minute. Here V80 : 7395 :: 60" or 1': 607 =75V 31:3 = . .
80 41.95987, or almost 42 vibrations in a minute.
153. In these propositions, the thread is supposed to be very fine, or of no sensible weight, and the ball very small, or all the matter united in one point; also, the length of the pendulum, is the distance from the point of suspension, or centre of motion, to this point, or centre of the small ball. But if the ball be large, or the string very thick, or the vibrating body be of any other figure; then the length of the pendulum is different, and is measured, from the centre of motion, not to the centre of magnitude of the body, but to such a point, as that if all the matter of the pendulum were collected into it, it would then vibrate in the same time as the compound pendulum; and this point is called the Centre of Oscillation; a point which will be treated of in what follows.
THE MECHANICAL POWERS, &€. 154. WEIGHT and Power, when opposed to each other, signify the body to be moved, and the body that moves it ; or the patient and agent. The power is the agent, which moves, or endeavours to move, the patient or weight.
155. Equilibrium, is an equality of action or force, between two or more powers or weights, acting against each other, by which they destroy each other's effects, and remain at rest,
156. Machine, or Engine, is any mechanical instrument contrived to move bodies. And it is composed of the mechanical powers.
157. Mechanical Powers, are certain simple instruments, commonly employed for raising greater weights, or overcoming greater resistances, than could be effected by the natural strength without them. These are usually accounted six in
number, viz. the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw.
158. Mechanics, is the science of forces, and the effects they produce, when applied to machines, in the motion of bodies.
159. Statics, is the science of weights, especially when considered in a state of equilibrium.
160. Centre of Motion, is the fixed point about which a body moves. And the Axis of Motion, is the fixed line about which it moves.
161. Centre of Gravity, is a certain point, on which a body being freely suspended, it will rest in any position.
OF THE LEVER.
162. A Lever is any inflexible rod, bar, or beam, which serves to raise weights, while it is supported at a point by a fulcrum or prop, which is the centre of motion. The lever is supposed to be void of gravity or weight, to render the demonstrations easier and simpler. There are three kinds of levers. 163. A Lever of the First
W:kind has the propc between the weight w and the power P.
And of this kind are balances, scales, crows, hand-spikes, scissors, pinchers, &c.
164. A Lever of the Second kind has the weight between the power and the prop. Such as oars, rud
с W ders, cutting knives that are fixed at one end, &c.
165. A Lever of the Third kind hasthe power between the weight and the prop. Such as tongs, the bones and muscles of animals, a man rearing a ladder, &c.
167. In all these instruments the power may be represented by a weight, which is its most natural measure, acting downward: but having its direction changed, when necessary, by means of a fixed pulley.
PROPOSITION XXXI. 168. When the Weight and Power keep the Lever in Equilibrio,
they are to each other Reciprocally as the Distances of their
W CB parallel to wo: Also, join co,
ID which will be the direction of the pressure on the prop c; for there cannot be an equilibrium unless the
F directions of the three forces all meet in, or tend to, the same point, as o.
Bli Then, because these three forces keep each other in equilibrio, they are proportional to the sides of the triangle ceo or Clo, drawn in the direction of those forces; therefore
P: W:: CF: Fo or CB. But, because of the parallels, the two triangles CDF, CEB are equiangular, therefore
CD: CE :: CF :CB. Hence, by equality,
P: W :: CD : ce. That is, each force is reciprocally proportional to the distance of its direction from the fulcrum.
And it will be found that this demonstration will serve for all the other kinds of levers, by drawing the lines as directed.
169. Corol. 1. When the angle A is = the angle w, then is CD : CE :: CW : CA ::P:w. Or when the two forces act perpendicularly on the lever, as two weights, &c; then, in case of an equilibrium, D coincides with w, and e with P; consequently then the above proportion becomes also'p:w:: CW:CA, or the distances of the two forces from the fulcrum, taken on the lever, are reciprocally proportional to those forces.
170. Corol. 2. If any force p be applied to a lever at A; its effect on the lever, to turn it about the centre of motion c; is as the length of the lever CA, and the sine of the angle of direction CAE. For the perp.ce is as ca * s. LA.
171. Corol. 3. Because the product of the extremes is equal to the product of the means, therefore the product of the power by the distance of its direction, is equal to the product of the weight by the distance of its direction.
That is, P X CE = W X CD.
172. Corol. 4. If the lever, with the weight and power fixed to it, be made to move about the centre c; the momentum of the power will be equal to the momentum of the weight; and their velocities will be in reciprocal proportion to each other. For the weight and power will describe circles whose radii are the distances CD, Ce; and since the circumferences or spaces described, are as the radii, and also as the velocities, therefore the velocities are as the radii cd; CE; and the momenta, which are as the masses and velocities, are as the masses and radii; that is, as Þ x ce and w x CD, which are equal by cor. 3.
173. Corol. 5. In a straight lever, kept in equilibrio by a weight and power acting perpendicularly; then, of these three, the power, weight, and pressure on the prop, any one is as the distance of the other two. 174. Corol. 6. If
DE several weights P, Q, R, s, act on a straight lever, and keep it in equilibrio; then the sum of the products on one side of the prop, will be equal to the sum on the other side; made by multiplying each weight by its distance ; namely, o X AC TR X BC = R X DC + 5 X EC.
For, the effect of each weight to turn the lever, is as the weight multiplied by its distance; and in the case of an equilibrium, the sums of the effects, or of the products on both sides, are equal.
175. Corol. 7. Because, when two weights & and are in equilibrio, 0 : R :: CD : CB;
Q therefore, by composition, etR:Q:: BD : CD,
and, é + RiR :: BD ; CB. VOL. II.