this excess. Let the moveable weight P, when placed at E, keep the lever at rest; then when W and P are suspended upon the lever, and the whole remains at rest, W sustains P, and also a weight which would support P when placed at E; therefore WX AC=P× DC+P×EC=P×DE; and since AC and P are invariable, WED; the graduation must, therefore, begin 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 at 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, one 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: ED. 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 when 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 in 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, 120. ON THE PULLEY. the 133. Def. A Pulley is a small wheel moveable about its centre, 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 changing its direction, is in the pressure pon the centre of motion. (Art. 140.) 137. In the single moveable pulley, whose strings are parallel, the power is to the weight as 1 to 2.† A string fixed at E, 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, an1 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 any numbe. Vide Art. 105. In this and the following propositions, the power and weight are supposed to be in equilibrio. of pulleys, and the parts of it between the pulleys are parallel, P: W :: 1: the number of strings at the lower block. Figures 37 and 38. Since the parallel parts, or strings at the lower block, are in the direction in which the weight acts, they exactly support the whole weight; also, the tension in every point of these strings is the same, otherwise the system would not be at rest, and consequently each of them sustains an equal weight; whence it follows that, if there be n strings, each sustains th part of the weight; therefore, P sustains th 1 n 1 n 142. Cor. If two systems of this kind be combined, in which there are m and a strings, respectively, at the lower blocks, P: W:: 1: mn. 143. In a system where eacn pulley hangs by a separate string, and the strings are parallel, P: W:: 1 that power of 2 whose index is the number of moveable pulleys. In this system, a string passes over the fixed pulley A, fig. 39, and under the moveable pulley B, and is fixed at E; another string is fixed at B, passes under the moveable pulley C, and is fixed at F; &c. in such a manner that the strings are parallel. Then, by art. 137, when there is an equilibrium, P the weight at B:: 1:2 the weight at B: the weight at C :: 1:2 the weight at C: the weight at D :: 1:2 &c. Comp. P: W:: 1:2 ×2×2×, &c. continued to as many factors as there are moveable pulleys; that is, when there are n such pulleys, P: W:1:2". 144. Cor. 1 The power and weight are wholly sustained at A, E, F, G, &c. which points sustain respectively, 2P, P, 2P, 4P, &c. 145. Cor. 2. When the strings are not parallel, P: W:: rad. : 2 cos. of the angle which the string makes with the direction in which the weight acts, in each case (art. 158). 146. In a system of n pulleys, each hanging by a separate string, where the strings are attached to the weight, as is represented in fig. 40, P: W:: 1 : 2′′—1. A string, fixed to the weight at F, passes over the pulley C, and is again fixed to the pulley B; another string, fixed at E, passes over the pulley B, and is fixed to the pulley A; &c. in such a manner that the strings are parallel. Then, if P be the power, the weight sustained by the string DA is P; also the pressure downwards upon A, or the weight which the string AB sustains, is 2 P (art. 140); therefore the string EB sustains 2P; &c. and the whole weight sustained is P+2P+4P+, &c. Hence P: W :: 1:1+2+4+, &c. to n terms: 1 2"-1. 147. Cor. 1. Both the power and the weight are sustained at H. 148. Cor. 2. When the strings are not parallel, the power in each case, is to the corresponding pressure upon the centre of the pulley; rad. : 2 cos. of the Z z angle made by the string with the direction in which the weight acts (art. 189). Also, by the resolution of forces, the power in each case, or pressure upon the former pulley, is to the weight it sustains:: rad. ; cos. of the angle made by the string with the direction in which the weight acts. ON THE INCLINED PLANE. 149. If a body act upon a perfectly hard and smooth plane, the effect produced upon the plane is in a direction perpendicular to its surface. Case 1. When the body acts perpendicularly upon the plane, its force is wholly effective in that direction; since there is no cause to prevent the effect, or to alter its direction. Case 2. When the direction in which the body acts is oblique to the plane, resolve its force into two, one parallel, and the other perpendicular, to the plane; the former of these can produce no effect upon the plane, because there is nothing to oppose it in the direction in which it acts (art. 19,, and the latter is wholly effective (by the first case); that is, the effect produced by the force is in a direction perpendicular to the plane. 150. Cor. The re-action of a plane is in a direction perpendicular to its surface (art. 20). 151. When a body is sustained upon a plane which is inclined to the horizon, P: W: the sine of the plane's inclination: the sine of the angle which the direction of the power makes with a perpendicular to the plane. Let BC, fig. 41, be parallel to the horizon, BA a plane inclined to it; P a body, sustained at any point upon the plane by a power acting in the direction PV. From P draw PC perpendicular to BA, meeting BC in C; and from C draw CV perpendicular to BC, meeting PV in V. Then the body P is kept at rest by three forces which act upon it at the same time; the power, in the direction PV; gravity, in the direction VC; and the re-action of the plane, in the direction CP (art. 150); these three forces are therefore properly represented by the three lines PV, VC, and CP (art. 59); or P: W:: PV: VC:: sin. PCV sin, VPC; and in the similar triangles APC, ABC (Euc. 8. 6), the angles ACP, and CBA are equal; therefore P: W :: sin. ABC : sin. VPC. 152. Cor. 1. When PV coincides with PA, or the power acts parallel to the plane P: W:: PA : AC :: AC: AB. 153. Cor. 2. When PV coincides with Pe, or the power acts parallel to the base, P: W: Pv : tC :: AC : CB; because the triangles PoC, ABC are similar. 154. Cor. 3. When PV is parallel to CV, the power sustains the whole weight. 1 1 155. Cor. 4. Since P: W: sin. ABC sin. VPC, by multiplying extremes and means, P × sin. VPC=Wx sin. ABC; and if W, and the sine of the ZABC be invariable, Pa ; therefore P is the least, when sin. VPC sin. VPC is the least, or sin. VPC the greatest; that is, when sin. VPC becomes the radius, or PV coincides with PA. Also, P is indefinitely great when sin. VPC vanishes; that is, when the power acts perpendicularly to the plane. 156. Cor. 5. If P and the ABC be given, WC sin. VPC; therefore W will be the greatest when sin. VPC is the greatest, that is, when PV coin. cides with PA. Also, W vanishes when the sin. VPC vanishes, or PV coin. cides with PC. 157. Cor. 6. The power: the pressure :: PV: PC:: sin. PCV: sin. PVC :: sin. ABC sin. PVC. : 158. Cor. 7. When the power acts parallel to the plane, the power the pressure:: PA: PC:: AC: BC: sin. B: cs. ZB. 159. Cor. 8. When the power acts parallel to the base, the power: the pres sure: Po: PC:: AC: AB. 160. Cor. 9. Px sin. PVC the pressure X sin ABC; and when P and the ZABC are given, the pressure sin. PVC; therefore, the pressure will be the greatest when PV is parallel to the base. 161. Cor. 10. When two sides of a triangle, taken in order, represent the quantities and di cctions of two forces which are sustained by a third, the remaining side, taken in the same order, will represent the quantity and direc tion of the third force (art. 58). Hence, if we suppose PV to revolve round P, when it falls between Pr, which is parallel to VC, and PE, the direction of gravity remaining unaltered, the direction of the re-action must be changed, or the body must be supposed to be sustained against the under surface of the plane. When it falls between PE and P produced, the direc tion of the power must be changed. And when it falls between P produced, and PC, the directions of both the power and re-action must be different from what they were supposed to be in the proof of the proposition; that is, the body must be sustained against the under surface of the plane, by a force which acts in the direction VP. 162. Cor. 11. If the weights P, W, fig. 42, sustain each other upon the planes AC, CB, which have a common altitude CD, by means of a string PCW which passes over the pulley C and is parallel to the planes, then P: W:: AC : BC. For, since the tension of the string is every where the same, the sustaining power, in each case, is the same; and calling this power x, P: x AC: CD (art. 152); x: W:: CD: CB; comp. P: W: AC: CB. ON THE WEDGE. 163. Def. A Wedge is a triangular prism; or a solid generated by the motion of a plane triangle parallel to itself, upon a straight line which passes through one of its angular points. Knives, swords, coulters, nails, &c. are instruments of this kind. The wedge is called isosceles or scalene, according as the generating triangle is isosceles or scalene. 164. If two equal forces act upon the sides of an isosceles wedge at equal angles of inclination, and a force act perpendicularly upon the back, they will keep the wedge at rest, when the force upon the back is to the sum of the forces upon the sides, as the product of the sine of half the vertical angle of the wedged the sine of the angle at which the directions of the forces are inc. ed to the sides, is to the square of radius. Let AVB fig. 43, represent a section of the wedge, made by a plane perpendicular to its sides; draw VC perpendicular to AB; DC, C, in the directions of the forces upon the sides; and CE, Ce, at right angles to AV, BV; join Ee, meeting CV in F. |