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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.
166. A Fourth kind is sometimes added, called the Bended Lever. As a hammer drawing à 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.
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, r: W:: CD: CE; where CD and CE are perpendicular to wo and so, the Directions of the two Weights, or the Weight and Power w and A.
FOR, draw cr parallel to Ao, and CB parallel to wo: Also join co, which will be the direction of the pressure on the prop c; for there cannot be an equilibrium unless the directions of the three forces all meet in, or tend to, the same point, as o. Then, because these three forces keep each other in equilibrio, they are proportional to the sides of the triangle CBO or cro, drawn in the direction of those forces; therefore
But, because of the parallels, the
Hence, by equality,
CD: CE :: CF : CB.
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 proportional 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 lever CA, and the sine of the angle of direction cae. For the perp. CE is as CAXSLA.
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 proTM duct of the weight by the distance of its direction.
That is, PXCE=W XCD.
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 cir cumferences 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 FX CE and wXCD, which are equal by cor. 3.
173. Corol. 5. 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.
equal to the sum on the
other side, made by multiplying each weight by its distance., namely,
For, the effect of each weight to turn the lever, is as the weight multiplied by its distance; and in the case of an equi librium, the sums of the effects, or of the products on both sides are equal.
175. Corol. 7. Because, when two weights & and R are in equilibrio, QR :: CD: CB; therefore,
by composition, Q+R: Q :: BD : CD, and, Q+R : R :: BD : GB,
That is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other.
176. On the foregoing prin-
1st. That the points of suspension of the scales and the centre of motion of the beam, A, B, C, should be in a straight line: 2d, That the arms AB, BC, be of an equal length: 3d, That the centre of gravity be in the centre of motion в, or a little below it: 4th, That they be in equilibrio when empty : 5th, That there be as little friction as possible at the centre B. A defect in any of these properties, makes the scales either imperfect or false. But it often happens that the one side of the beam is made shorter than the other, and the defect covered by making that scale the heavier, by which means the scales hang in equilibrio when empty; but when they are charged with any weights, so as to be still in equilibrio, those weights are not equal; but the deceit will be detected by changing the weights to the contrary sides, for then the equi librium will be immediately destroyed.
177. To find the true weight of any body by such a false balance:First weigh the body in one scale, and afterwards weigh it in the other; then the mean proportional between these two weights, will be the true weight required. For, if any body b weigh w pounds or ounces in the scale D, and only wpounds or ounces in the scale E: then we have these two equations, namely, AB. b=BC. w.
and BC. b⇒AB. W;
the product of the two is AB. BC . b2.
the mean proportional, which is the true weight of the body b.
178. The Roman Statera, or Steelyard, is also a lever, but of unequal brachia or arms, so contrived, that one weight only may serve to weigh a great many, by sliding it back
ward and forward, to different distances, on the longer arm of the lever; and it is thus constructed :
Let AB be the steelyard, and c its centre of motion, whence the divisions must commence if the two arms just balance each other: if not, slide the constant moveable weight 1 along from B towards c, till it just balance the other end without a weight, and there make a notch in the beam, marking it with a cipher 0. Then hang on at a a weight w equal to 1, and slide 1 back towards в till they balance each other; there notch the beam, and mark it with 1. Then make the weight w double of 1, and sliding 1 back to balance it, there mark it with 2. Do the same at 3, 4, 5, &c. by making w equal to 3, 4, 5, &c. times i; and the beam is finished. Then to find the weight of any body b by the steelyard; take off the weight w, and hang on the body b at a then slide the weight 1 backward and forward till it just balance the body b, which suppose to be at the number 5; then is b equal to 5 times the weight of 1. So, if 1 be one pound, then bis 5 pounds; but if I be 2 pounds, then b is 10 pounds; and so on.
179. In the Wheel-and-Axle; the Weight and Power will be in Equilibrio, when the Power P is to the Weight w, Reciprocally as the Radii of the Circles where they act; that is, as the Radius of the Axle ca, where the Weight hangs, to the Radius of the Wheel CB, where the Power acts. That is,
P: W CA: CB.
HERE the cord, by which the power Facts, goes about