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in the same time that it described CD before impact; also these spaces are uniformly described ; consequently, the velocity before inpact: the velocity after :: CD: CE :: radius : sin. 2 CDE :: radius ; cos. < CDA.
96. Cor. If CD be taken to express the direction and absolute quaotity of oblique force with which the body impinges on the plane AB, it may, in the very same manner be shown, that ED will express the direct force, or energy of the stroke upon that plane, and we shall have
oblique force : direct force :: CD : DE :: radius : sin. angle of incidence.
97. If a perfectly elastic body impinge upon an immoveable plane AB, fig. 13, in the direction CD, it will be reflected from it in the direction DF, which makes, with DB, the angle BDF, equal to the angle ADC.
Let CD represent the motion of the impinging body ; draw CF parallel, and DE perpendicular, to AB; make EF=CE, and join DF. Then the whole motion may be resolved into the two CE, ED, of which CE is employed in carrying the body parallel to the plane, and must therefore remain after the impact ; and ED carries the body in the direction ED, perpendicular to the plane ; and since the plane is immoveable, this motion will be destroyed during the compression, and an equal motion will be generated in the opposite direction by the force of elasticity. Hence, it appears that the body at the point D has two motions, one of which would carry it uniformly from D to E, and the other from E to F in the same time, viz. in the time in which it described CD before the impact; it will, therefore, describe DF in that time (art. 47). Again, in the triangles CDE, EDF, CE=EF, the side ED is common, and the 2 CED= ZDEF; therefore, the 2CDE = Z FDE ; hence, the Z CDA= Z FDB.
98. Cor. Since CD=DF, and these are spaces nniformly described in equal times, before and after the impact, the velocity of the body atter the reflection is equal to its velocity before the incidence.
99. To determine the motions after the impact, in two bodies which strike one another obliquely.
Let the two bodies A and B, fig. 14, move in the oblique directions AK, BK, and strike each other at K, with velocities which are in proportion to the lines AK, CK; to find their motions after the impact. Let CKH represent the plane in which the bodies touch in the point of concourse ; to this plane draw the perpendiculars AC, BD, and complete the rectangles CE, DF. Then the motion in AK is resolved into the two AC, CK ; and the motion in BK is resolved into the two BD, DK; of which the antecedents AC, BD, are the velccities with which they directly meet, and the consequents CK, DK, are parallel ; therefore, by these the bodies do not impinge on each other, and consequently the motions, according to these directions, will not be changed by the impulse ; so that the velocities with which the bodies meet are AC and BD, or their equals EK, FK. The motions of the bodies A, B, directly striking each other with the velocities EK, FK, will be determined by articles 76, 87, &c. according as the bodies are elastic or non-elastic ; which being done, let KG be the velocity so determined, of one of them, as A ; and since there ernains also in the body a force of moving in the direction parallel to AE, with a velocity as AE, make KH=AE, and complete the rectangle GH. Then the two motions in KH, and KG, or HI, are compounded into the diagonal KI, which, therefore, will be the path and velocity of the body A after the stroke. And after the same manner may the motion of the other body B be determined after the inipact.
İf the elasticity of the bodies be imperfect in any given degree, then the quantity of the corresponding lines must be diminished in the same proportion.
ON THE MECHANICAL POWERS. 100. The mechanical powers are the most simple instruments used for the purpose of supporting weights, or communicating motion to bodies, and by the combination of which, all machines, however complicated, are constructed.
These powers are six in number, viz. the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw.
Before we enter upon a particular description of these instruments and the calculation of their effects, it is necessary to premise, that when any forces are applied to them, they are themselves supposed to be at rest, and consequently that they are either without weight, or that the parts are so adjusted as to sustain each other. They are also supposed to be perfectly smooth ; no allowance being made for the effects of adhesion.
When two forces act upon each other by means of any machine, one of them is, for the sake of distinction, called the power, and the other the weight.
On the Lever. 101. Def. The Lever is an inflexible rod, moveable upon a point which is called the fulcrum, or centre of motion.
The power and weight are supposed to act in the plane in which the lever is moveable round the fulcrum, and tend to turn it in opposite directions.
Ar. 1. If two weights balance each other upon a straight lever, the prese sure upon the fulcrum is equal to the sum of the weights, whatever be the length of the lever.
Xx. 2. If a weight be supported upon a lever which rests on two falcrums, the pressure upon the fulcrums is egnal to the whole weight.
Ar. 3. Equal forces, acting perpendicularly at the extremities of equal armas of a lever, exert the same effort to turn the lever round.
102. If two equal weights act perpendicularly upon a straight lever,
The effect produced by the gravity of the lever is not taken into considevation, unless it be expressly mentionedo,
the effort to put it in motion, rouna any fulcrum, will be the same as if they acted together at the middle point between them.
Let A and B be two equal weights, acting perpendicularly upon the lever FB, whose fulcrum is F, fig. 15. Bisect AB in C, make CE=CF, and at E suppose another fulcrum to be placed.
Then, since the two weights A and B are supported by E and F, and these fulcrums are similarly situated with respect to the weights, each sustains an equal pressure ; and, therefore, the weight sustained by E is equal to half the sum of the weights. Now let the weights A and B be placed at C, the middle point between A and B, and consequently the middle point between E and F; then, since É and F support the whole weight C, and are similarly situated with respect to it, the fulcrum E supports half the weight; that is, the pressure upon E is the same, whether the weights are placed at A and B, or collected in C, the middle point between them; and, therefore, the effort to put the lever in motion round F, is the same on either supposition.
103. Cor. If a weight be formed into a cylinder AB ( fig. 16) which is every where of the same density, and placed parallel to the horizon, the effort of any part AD, to put the whole in motion round C, is the same as if this part were collected at E, the middle point of AD.
For the weight AD may be supposed to consist of pairs of equal weights, equally distant from the middle point.
What is here affirmed of weights, is true of any forces which are propor. tional to the weights, and act in the same directions.
104. Two weights, or two forces, acting perpendicularly upon a straight lever, will balance each other, when they are reciprocally proportional to their distances from the fulcrum.
Case 1. When the weights act on contrary sides of the fulcrum.
Let x and y be the two weights, and let them be formed into the су linder AB, fig. 16, which is every where of the same density. Bisect AB in C, then this cylinder will balance itself upon the fulcrum C (art. 103). Divide AB into two parts in D, so that AD: DB ::: y, and the weights of AD and DB will be respectively x and y; bisect AD in E and DB in F; then, since AD and DB keep the lever at rest, they will keep it at rest when they are collected at E and F (art. 103); that is, x, when placed at E, will balance y, when placed
AD BD AB-BD AB-AD at F; and x:y :: AD : BD ::
2. CB-BF: AC-AE :: CF : CE.
Case 2. When the two forces act on the same side of the centre of motion.
Let AB, fig. 17, be a lever whose fulcrum is C, A and B two weights acting perpendicularly upon it; and let A : B :: BC : AC, then these weights will balance each other, as appears by the former case. Now suppose a power sufficient to sustain a weight equal to the sum of the weights A and B, to be applied at C, in a direction opposite to that in which the weights act; then will this power supply the place of the fulcrum (art. 101, ar. 1); and the centre of motion
may be conceived to be at A or B. Let B be the centre of motion 3 dien we have a straight lever whose centre of motion is B, and the two forces A and A+ B, acting perpendicularly upon it at the points A and C, sustain each other ; also, À : B :: BC : AC; therefore A : A+B :: BC : BA.
105. Cor. 1. If two weights, or two forces, acting perpendicularly on the arms of a straiglit lever, keep each other in equilibrio, they are inversely as their distances from the centre of motion.
For the weights will balance when they are in that proportion, and if the proportion be altered by increasing or diminishing one of the weights, its effort to turn the lever round will be altered, or the equilibrium will be destroyed.
106. Cor. 2. Since A:B :: BC: AC when there is an equilibrium upon the lever AB, whose fulcrum is C, by multiplying extremes and means, AXACBXBC.
107. Cor. 3. When the power and weight act on the same side of the fulcrum, and keep each other in equilibrio, the weight sustained by the fuleram is equal to the difference between the power and the weight.
108. Cor. 4. In the common balance, the arms of the lever are equal; consequently, the power and weight, or two weights, wbich sustain each other, are equal. In the false balance, one arm is longer than the other; therefore the weight, which is suspended at this arm, is proportionally less than the wcight which it sustains at the other.
109. Cor. 5. If the same body be weighed at the two ends of a false balance, its true weight is a mean proportional between the apparent weights.
Call the true weight x, and the apparent weights, when it is suspended at A and B, d and b respectively; then a : .::AC : BC, and x:b:: AC: BC; therefore a : x :: 3:1.
110. Cor. 6. If a weight C, fig. 18, be placed upon a lever which is supported upon two props A and B in an horizontal position, the pressure upon A : the pressure upon B :: BC: AC.
For if B be conceived to be the fulcrum, we have this proportion, the weight sustained by A : the weight C :: BC: AB; in the same manner, if A be considered as the fuļcrum, then the weight C : the weigbt sustained by B :: AB : CA ; therefore, ex æquo, the weiglit sustained by A : the weight sustained by B :: BC:AC.
111. Cor. 7. If a given weight P, fig. 19, be moved along the graduated arm of a straight lever, the weight w, which it will balance at A, is proportional to CD, the distance at which the given weight acts.
When there is an equilibrium, WXAC=PX DC (art. 106); and AC and P are invariable; therefore wa DC.
112. If two forces, acting upon the arms of any lever, keep it at rest, they are to each other inversely as the perpendiculars drawn from the centre of motion to the directions in which the forces act.
Case 1, Let two forces, A and B, fig. 20, act perpendicularly upon the arms CA, CB, of the lever ACB whose fulcrum is C, and keep each other at rest. Produce BC to D, and make CD=CA; then the effort of A to move the lever round C, will be the saine, whether it be supposed to act perpendicularly at the extremity of the arm CA, or CD (art. 101, ar. 3); and on the latter suppositior, since there is an equilibrium, A :B :: CB : CD (art. 105); therefore A :B :: CB : CA.
Case 2. When the directions AD, BH, fig. 21, in which the forces act, are not perpendicular to the arms, take AD and BH, to represent