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forces ; draw CM and CN at right angles to those directions; also draw AF perpendicular, and DF parallel to AC, and complete the
parallelogram GF; then the force AD is equivalent to the two AF, AG, of which AG acts in the direction of the arm, and therefore can have no effect in causing, or preventing any angular motion in the lever about C. Let BH be resolved, in the same manner, into the two BI, BK, of which BI is perpendicular to, and BK in the direction of the arm CB; then BK will have no effect in causing, or preventing any angular motion in the lever about C; and since the lever is kept at rest, AF and BI, which produce this effect, and act perpendicularly upon the arms, are to each other, by the first case, inversely as the arms; that is, AF: BI :: CB : CA, or AFX CA=BI X CB. Also, in the similar triangles ADF, ACM, AF : AD :: CM : CA, and AFX CA=AD CM; in the same manner, BIX CB=BHXCN; therefore ADX CM=BHX CN, and AD : BH :: CN : CM.
113. Cor. 1. Let a body IK, fig. 22, be moveable about the centre C, and two forces act upon it at A and B, in the directions AD, BH, which coincide with the plane ACB; join AC, CB; then this body may be considered as a lever ACB, and drawing the perpendiculars CM, CN, there will be an equilibrium, when the force acting at A: the force acting at B :: CN : CM.
114. Cor. 2. The effort of the force A, to turn the lever round, is the same, at whatever point in the direction MD it is applied ; because the perpendicular CM remains the same.
CA X sin. CAM 115. Cor. 3. Since CA: CM :: rad. : sin. CAM, CM =
rad. CB X sin, CBN and, in the same manner, CN
therefore, when there is an
Casin. CBN CAasin. CAM equilibrium, the power at A: the weight at B::
rad. :: CB X sip. CBN : CA X sin. CAM. 116. Cor. 4. If the lever
ABC fig. 23, be straight, and the directions AD, BH, parallel, A : B :: BC:
AC; because, in this case, sin. CAM=sin. CBH. Hence, also, AXAC = BXBC. . 117. Cor. 5. If two weights balance each other upon a straight lever in any one position, they will balance each other in any other position of the lever; for the weights act in parallel directions, and the arms of the lever are in variable.
118. Cor. 6. If a man, balanced in a common pair of scales, press upwards by means of a rod, against any point in the beam, except that from which the scale is suspended, he will preponderate. Let the action upwards take place at D,
fig. 24, then the scale, by the re-action downwards, will be brought into the situation E; and the effect will be the same as if DA, AE, DE, constituted one mass ; that is, drawing EF perpen. dicular to CA produced, as if the scale were applied at F (art. 114); consequently the weight, necessary to maintain the equilibrium, is greater than if the scale were suffered to hang freely from A, in the proportion of CF: CA.
119. Cor. 7. Let AD fig. 23, represent a wlieel, bearing a weight at its centre C; AB an obstacle over which it is to be moved by a force acting in the direction CE; join CA, draw CD perpendicular to the horizon, and from A draw AG, AF, at right angles to CE, CD. Then CA may be considered as a lever whose centre of motion is A, CD the direction in which the weight acts, and CE the direction in which the power is applied; and there is an equilibrium on this lever when the power : the weight :: AF :: AG.
Supposing the wheel, the weight, and the obstacle given, the power is
least when AG is the greatest; that is, when CE is perpendicular to CA, JE parallel to the tangent at A.
120. Cor. 8. Let two forces acting in the directions AD, BH, fig. 26, npon the arms of the lever ACB, keep each other in equilibrio ; produce DA and HBm: till they meet in P ; join CP, and draw CL parallel to PB; then will PL, LC represent the two forces, and PC the pressure upon the fulcrum.
For, if PC be made the radins, CM and CN are the sides of the angles CPM, CPN, or CPL, PCL; and PL:LC :: sin. PCL: sin. LPC::CN: CM; therefore PL, LC, represent the quantities and directions of the two forces, which may be supposed to be applied at P (art. 114), and which are sustained by the re-action of the fulcrum ; consequently, CP represents the quantity and direction of that re-action (wrt. 58), or PC represents the pressure upou the fulcrum.
121. In a combination of straight levers, AB, CD, figs. 27, 28, whose centres of motion are E and F, if they act perpendicularly upon each other, and the directions in which the power and weight are applied be also perpendicular to the arms, there is an equilibrium when P:W:: EB XFD: EA X FC.
For the power at A : the weight at B, or C :: EB : EA; and the weight at C: the weight at D :: FD : FC; therefore, P: W :: EB X FD : EA X FC.
By the same method we may find the proportion between the power and the weight, when there is an equilibrium, in any other combination of levers.
122, Cor. If E and F be considered as the power and weight, A and D the centres of motion, we have, as before, E:F:: FDXBA:AEXCD. Hence the pressure upon E. the pressure upon F :: FDX BA : AEXCD.
123. Any weights will keep each other in equilibrio on the arms of a straight lever, when the products, which arise from multiplying each weight by its distance from the fulcrum, are equal on cach side of the fulcrum.
The weights A, B, D, and EF, fig. 29, will balance each other upon the lever AF whose fulcrum is C, if A XAC+B x BC+Dx DC= EXEC+Fx FC.
In CF take any point X, and let the weights r, s, t, placed at X, balance respectively, A, B, D; then AX AC=rxXC; B x BC= sx XC ; D DC=tx XC (art. 116); or, A X AC+B x BC+Dx DC =rts+ixXC. In the same manner, let p and q, placed at Y, balance respectively, E and F; then p+-9xYC=EX EC+Fx FC; but by the supposition AXAC+B X BC +DxDC=EXEC+Fx FC; therefore r+stix XC=p+q* YC, and the weights r, s, t, placed at X, balance the weights P. q, placed at Y; also A, B, D, balance the former weights, and E, F, the latter ; consequently A, B, D, will balance E, and F.
124. Cor. 1. If the weights do not act in parallel directions, instead of the distances, we must substiinte the perpendiculars, drawn from the centre of motion, upon the directious. (Art. 112.)
125. Cor. 2. In art. 111, the lever is snpposed to be without weight, or the arms AC, CD to balance each other. In the formation of the common steel yarch fig. 30, the longer arm CB is heavier than CA, and allowance must be made for
this excess. Let the moveable weight P, when placed at E, keep the lever at rest ; then when W and P are suspeaded upon the lever, and the whole remains at rest, W spstains P, and also a weight which_wonld support P when placed at E; therefore WXAC=PxDC+PXEC=PX DE; and since AC and P are invariable, WOCED; the graduation must, therefore, begio from E; and if P, when placed at F, support a weight of one pound at A, take FG, GD, &c. equal to each other, and to EF, and when P is placed ai G it will support two pounds; when at D it will support three pounds, &c.
ON THE WHEEL AND AXLE. 126 The wheel and axle consists of two parts, a cylinder AB, fig. 31, moveable about its axis CD, and a circle EF so attached to the cylinder that the axis CD passes through its centre, and is perpendicular to its plane.
The power is applied at the circumference of the wheel, usually in the direction of a tangent to it, and the weight is raised by a rope which winds round the axle in a plane at right angles to the axis.
127. There is an equilibrium upon the wheel and axle, when the power is to the weight, as the radius of the axle to the radius of the wheel.
The effort of the power to turn the machine round the axis, must be the same at whatever point in the axle the wheel is fixed ; suppose it to be removed, and placed in such a situation that the power and weight may act in the same plane, and let CA, CB, fig. 32, be the radii of the wheel and axle, at the extremities of which the power and weight act; then the machine becomes a lever ACB, whose centre of motion is C; and since the radii CA, CB, are at right angles to AP and BW, we have P:W:: CB : CA (art. 113).
128. Cor. 1. If the power act in the direction Ap, draw CE perpendicular to Ap, and there will be an equilibrium when P: W :: CB : CE (art. 113.)
The same conclusion may also be obtained by resolving the power into two, ove perpendicular to AC, and the other parallel to it.
129. Cor. 2. If 2R be the thickness of the ropes by which the power and weight act, there will be an equilibrium when P: W:: CB+R: CA+R, since the power and weight must be supposed to be applied in the axes of the ropes.
The ratio of the power to the weight is greater in this case than the former; for if any quantity be added to the terms of a ratio of less inequality, that ratio is increased.
130. Cor. 3. If the plane of the wheel be inclined to the axle at the angle EOD, fig. 33, draw ED perpendicular to CD; and considering the wheel and axle as one mass, there is an equilibrium when P:W :: the radius of the axle :
131. Cor. 4. In a combination of wheels and axles, where the circumfer. ence of the first axle is applied to the circumference of the second wheel, by means of a string, or by tooth and pinion, and the second axle to the third wheel, &c. there is an equilibrium wben P: W:: the product of the radii of all the axles : the product of the radii of all the wheels. (Art. 121).
132. Cor. 5. When the power and weight act im parallel directions, and on opposite sides of the axis, the pressure upon the axis is equal to their sum; and when they act on the same side, to their difference. In other cases the pressure may be estimated by art, 190.
ON THE PULLEY. 133. Def. A Pulley is a small wheel moveable about its centre, the circumference of which a groove is formed to admit a rope or flexible chain.
The pulley is said to be fixed or moveable according as the centre of motion is fixed or moveable.
134. In the single fixed pulley, there is an equilibrium when the power and weight are equal.
Let a power and weight P, W, fig. 34, equal to each other, act by means of a perfectly flexible rope PDW which passes over the fixed pulley ADB ; then, whatever force is exerted at D in the direction DAP, by the power, an equal force is exerted by the weight in the direction DBW; these forces will therefore keep each other at rest.
135. Cor. 1. Conversely, when there is an equilibrium, the power and weight are equal."
136. Cor. 2. The proposition is true in whatever direction the power is applied; the only alteration made, by changiog its direction, is in the pressure upon the centre of motion. (Art. 140.)
137. In the single moveable pulley, whose strings are parallel, the power is to the weight as I lo 2.4
A string fixed at £, fig. 35, passes under the moveable pulley A, and over the fixed pulley B; the weight is annexed to the centre of the pulley A, and the power is applied at P. Then since the strings EA, BÁ are in the direction in which the weight acts, they exactly sustain it; and they are equally stretched in every point, therefore they sustain it equally between them; or each sustains half the weight. Also, whatever weight AB sustains, P sustains (art. 135) therefore P:W::1: 2.
138. In general, in the single moveable pulley, the power is to the weight, as radius to twice the co-sine of the angle which either string makes with the direction in which the weight acts.
Let AW, fig. 36, be the direction in which the weight acts ; produce BD till it meets AW in C, from A draw AD at right angles to AC, meeting BC in D; then if CD be taken to represent the power at P, or the power which acts in the direction DB, CA will represent that part of it which is effective in sustaining the weight, and AD will be counteracted by an equal and opposite force, arising from the tension of the string Ce; also, the two strings are equally effective in sustaining the weight; therefore 2AC will represent the whole weight sustained ; consequently, P:W:: CD : 2AC :: rad. : 2 cos. DCĂ.
139. Cor. 1. If the figure be inverted, and E and B be considered as a power and weight which sustain each other upon the fixed pulley A, W is the pressure upon the centre of motion ; consequently, the power : the pressure .: radius : 2 cos. DCA.
140. Cor. 2. When the strings are parallel, the angle DCA vanishes, an' its co-sine becomes the radius ; in this case, the power : the pressure :: 1 : 2. 141. In a system where the same string passes round
• Vide Art. 105. + In this and the following propositions, the power and weight are supposed to be in equilibrio.'