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149. Corot. '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 falliog 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 = 3.1416, and I the length of the pendulum, also & the space falten by a heavy body in ?" of time. then v 5: V}::1": v the time of falling through it,
25 theref. 1:1: :v ty
which therefore is the time of
28 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 p ; therefore the time of vibration in the small arc of a circle, is nearly equal to the time of vibra. tion in the cycloidal arc; consequently the time of vibration 'in a small circular, arc is equal to to w where l is the radius of the circle,
151. So that, if one of these, ş orl, 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 pen du. lum. Or, if the length of the second pendulum be observed by experiment, which is the easier way, this theorem will give s 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
395 written for in theorem, it gives pv
25 nila in? X 391 = 193.07 inches to 164' 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 pendulum that shall
number of vibrations in a given time. Or, the number of vibrations that shall be made by a pendulum of a given length. Thus, suppose it were required to 6nd the length of pendulum
a half-seconds pendulum. or a quarter-seconds i that is, a pendulum, to vibrate twice in a second,
Then since the time of vibration is as the square root of the length,
or 4 times in a second.
therefore 1:1:: ~ 39:11,
39 1:1:: 39:
= 91 inches nearly, the length of the half-seconds pendulum. Again 1 : 7 :: 394 : 24 inches, 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
29 V 80 : V 393 :: 60" or 1': 60 V- = 71 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, &c. 154. WEIGHT and Power, when opposed to each other, signify the body to be moved, and the body thai moyes 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 effecis, 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 Pully, 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.
And of this
Į66. A Fourth kind is sometimes added, called the Bended W Lever. As a hammer drawing
P a nail.
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 Lines of Direction from the Prop. That is, Piw::CD : CE ; where cd and ce are ferpendicular to wo and 10. the Directions of the two Weights, or the Weight and Power w and A.
For, draw cr parallel to A0, and CB parallel to wo : Also join co, which will be the direction of the
D pressure on the propc; for their E cannot be an equilibrium unless the directions of the three forces all meet in, or tend to, the same point, as o.
BY Then, because these three forces keep each other in equilibrio, they are proportional to the sides of the triangle ceo or cro, drawn in the direction of those 'forces ; therefore
P: W::cr: ro 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 js 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 a wiih P ; consequently then the above proportion becomes also p:W:: CW: GA, 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 - Jever ca, and the sine of the angle of direction CAE.
For the perp. ce is as ca X s. 6 A. 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 oiher. For the weight and power will describe cir. cles 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 P X ce and w X CD, which are equal by cor. 3.
173. Corol. 5. Io a straight lever, kept in equilibrio by a
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
D two weights Q and R are in
A equilibrio, Q : R : : CD
: CB ;