« ForrigeFortsett »
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 1
1 strings, each sustains th part of the weight; therefore, P sustains th
part of the weight, or P:W:: :1:: 1:n.
142. Cor. If two systems of this kind be combined, in which there are m and n strings, respectively, at thic lower blocks, P:W:: 1 : mn.
143. In a system where eacn pulley hangs ly a separate string, and the strings are parallel, P :W:: 1 : that power of 2 whose inder 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 B: the weight at C:: 1:2 the weight at C: the weight at D :: 1:2
&c. Consp. P:W:: 1:2 x2x2x, &c. continued to as many factors as there are nioveable 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 sastain respectively, 2P, P, 2P, 4P, &c.
145. Cor. 2. Wben the strings are not parallel, P:W:: rad. : ? cos. of the angle wbich 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. ton terms :: 1:2" — ).
117. ('or, 1. Both the power and the weighit 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 ceutre of the pulley :: rad. : 2 cos. of the
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 wbich 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 directioo 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 Po, or the power acts parallel to the base, P:W:: Pu: «C :: AC:CB; because the triangles PoC, ABC are similar.
154. Cor. 3. When PV is parallel to CV, the power sustains the whole weight.
155. Cor. 4. Since P:W:: sin. ABC : sin. VPC, by multiplying extremes and means, P x sin. VPC-WX sin. ABC; and if w, and the side of the ABC be invariable, Pa
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, W'OC sin. VPC ; therefore W will be the greatest wien 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:8.9. ZB.
159. Cor. 8. When the power acts parallel to the base, the power : tue pres. dure :: Po: PC :: AC: AB.
160. Cor. 9. Px sin. PVC=the pressure x sin ABC; and when P and the LABC are given, the pressure a 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 directions 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 dy must be supposed to be sustained against the under surface of the piane. When it falls between PE and xP produced, the direc. iion of the power must be changed. And when it falls between xP 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. 19, sustain each other npon the planes AC, CB, which have a common altinde CD, by ineans of a string PCW which passes over the pulley C and is parallel to the planes, then P: W:: AC: BC.
For, siuce the tension of the string is every where the same, the sustaining power, in each case, is the same; and calling this power it,
P: x :: AC : CD (art. 132);
I: 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 u'edge 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 lack is to the sum of the forces upon the sides, as the product of the sine of half the vertical angle of the wedge. d the sine of the angle at which the directions of the forces are int. ved 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, ac, in the directions of the forces upon the sides ; and CE, Ce, ar right angles to AV, BV ; join Ee, meeting CV in F
Then, in the triangles VCA VCB, since the angles VCA, CAV, are respectively equal to VCB, VBC, and VC is common to both, AC=CB, and the ZCVA= ZCVB. Again, in the triangles ACD, BCd, the angles DAC, CDA, are equal to the angles CBd, BdC, and AC=BC; therefore, DC=dC. In the same manner it may be shown that CF=Ce, and AE=Be; hence the sides AV, BV, of the triangle AVB, are cut proportionally in E and e; therefore Ee is parallel to AB (Euc. 2. 6), or perpendicular to CV; also, since CE=Ce, and CF is common to the right-angled triangles CEF, CeF, we have EF=eF (Euc. 47. 1).
Now since DC and dC are equal, and in the directions of the forces upon the sides, they will represent them ; resolve DC into two, DE, EC, of which DE produces no effect upon the wedge, and EC, which is effective (art. 149), does not wholly oppose the power, or force upon the back ;
resolve EC therefore into two, EF, parallel to the back, and FC perpendicular to it, the latter of which is the only force which opposes the power. In the same manner it appears that eF, FC are the only effective parts of dC, of which FC opposes the power, and eF is counteracted by the equal and opposite force EF ; bence if 2CF represent the power, the wedge will be kept at rest ;* that is, when the force upon the back : the sum of the resistances upon the sides :: 2CF : DC+dC :: 2 CF : 2DC :: CF : DC; an
CF: CE :: sin. CEF : rad. :: sin. CVE: rad.
CE: DC :: sin. CDE: rad.
165. Cor. 1. The forces do not sustain cach other, because the parts DE, de, are not counteructed.
166. Cor. 2. If the resistances art perpendicularly upon the sides of the wedge, the angle CDE becomes a rigtii angle, and P: the sum of the resistances :: sin. CVC X rad, : rad.2 :: sin. CVE: rad. :: AC: AV.
167. Cor. 3. If the direction of the resistances be perpendicular to the back, the angle CDE= Z CVE, and P: the sum of the resistances : sìn. CVE)* : rad. :: AC? : AV.
168. Cor. 4. When the resistances act parallel to the back, sin. CDA= sin. CAV, and P : the sun of the resistances :: sin. CVA X sin CAV: rad.)* :: CA XCV: AV2 :: CEXAVF: AV! :: CE: AV.
169. Cor. 5. In the demonstration of the proposition it has been supposed that the sides of the wedge are perfectly smooth; if on account of the friction, or by any other means, the resistances are wholly effective, join Dd, fig. 44, which will cut CV at right angles in y, and resolve DC, dC into Dy, yc, dy, yc, of which Dy and dy destroy each other, and 2yC sustains the power. Hence, the power : to the sum of the resistances :: 2yC: 2DC :: yC: DC :: sin. CDy or DCA: rad.
170. Cor. 6. If Ee cnt DC and dC in x and 2, the force, xC, zl', when wholly effective, and the forces DC, dC, acting upon smooth surfaces, will susu tain the same power 2CF.
171. Cor. 7. If from any point p in the side AV, PC be drawn, and the
The directions of the three forces must ineet in a point, utherwise a rotatory motion will be given to the wedge.
+ By similar triangles, CE: CA :: CV ; AV; therefore CEXAV -- CAX
resistance upon the side be represented by it, the effect upon the wedge will be the same as before ; the only difference will be in the part PE, which is ineffective.
172. Cor. 8. If DC be taken to represent the resistance on one side, and pC, greater or less than dC, represent the resistance on the other, the wedge cannot be kept at rest by a power acting upon the back; because, on this sny position, the forces which are parallel to the back are unequal.
Tliis proposition and its corollaries have been deduced from the actual resoJution of the forces, for the purpose of showing what parts are lost, or de. sloyed by their opposition to each other; the same couclusions may, however, be very concisely and easily obtained from art. 175.
173. When three forces, acting perpendicularly upon the sides of a scalene wedge, keep each other in equilibrio, they are proportional to those sides.
Let GI, HI, DI, the directions of the forces, meet in I, fig. 45; then, sirce the forces keep each other at rest, they are proportional to the th'ee sides of a triangle, which are respectively perpendicular to those directions (art. 58); that is, to the three sides of the wedge.
174. Cor. If the lines of direction, passing through the points of impact, do not meet in a point, the wedge will have a rotatory motion communicated to it; and this motion will be round the centre of gravity of the wedge as will be shown hereafter.
175. Cor. 2. When the directions of the forces are not perpendicular to the siele., the effective parts must be found, and there will be an equilibrium when 110: e parts are to each other as the sides of the wedge.
ON THE SCREW.. 176. Def. The
rew is a niechanical power, which may be conceived to be generated in the following manner :
I et a solid and a hollow cyliuder of equal diameters be taken, and let ABC, fig. 46, be a right-angled plane triangle, whose base BC is equal to tie circumference of the solid cylinder ; apply the triangle to the coniex surface of this cylinder, in such a manner that the base BC may coincide with the circumference of the base of the cylinder, and BA will form a spiral thread on its surface. By applying to the cylinder, triangles in succession, similar and equal to ABC, in such a manner, that their bases may be parallel to BC, the spiral thread may be continued : and, supposing this thread to have thickness, or the cylinder to be protuberant where it falls, the external screw will be forned, in which the distance between two contiguous threads, measured in a direction parallel to the axis of the cylinder, is AC. Again, let ihe triangles be applied in the same manner to the concave surface of the hollow cylinder, and where the thread falls let a groove be made, and the internal screw will be formed. The two screws being thus exactly adapted to each other, the solid or hollow cylinder, fig. 47, as die case requires, may be moved round the common axis, by a lever perpendicular to that axis ; and a motion will be produced in the direcilvil if the axis, by means of the spiral thread.
177. When there is an equilibrium upon the screw, P:W:: the dislunce between two contiguous threads, measured in a direction