122. To determine the true weight of a body by means of a false balance. In a false balance the arms of the beam are of unequal length. Let a=the length of the long arm; w the true weight of the body; Pthe weight of the body when placed in the scale attached to the long arm; p its weight when placed in opposite scale. Now as w is balanced in both cases, we have by the principle of the lever, wxa=pxb, and wxb=pxa, multiplying these equations together w2ab=ppab, .. w= √pp, which give the true weight required. 123. To find the force necessary to draw a carriage over a small obstacle. Let o be the centre of the wheel, F the obstacle, w the weight of the carriage acting through the axis o, Op the direction of the traction P. From F draw FA, FB, perpendicular to the direction of the forces P and w. Now, in order that the wheel may turn over the obstacle, it must turn round F as a centre of motion; F B D hence we have, by the quality of moments, Art. 117., PXFAWX FB, W Here P will be a minimum when FA is a maximum, that is, when FA equals FO or when OP is perpendicular to Fo. Put roF the radius of the wheel, and h=DB the weight of the obstacle, and <=/FOA; then the preceding condition becomes because h is small as compared with r. Hence it appears that the power necessary to draw the carriage over an obstacle (other things being the same) varies inversely as the square root of the radius of the wheel. 124. The principle of the equality of work as applied to the lever. (1). Let pw be a straight lever acted upon by the pressures P and w applied perpendicularly to the lever. Now when the lever is moved to the position pw, the pressure P has moved over the arc Pp, while w has moved over ww; P Fig. 68. .. Work of P=PX arc Pp, and work of w=wx arc ww; but by the principle of the lever, Art. 114., W (2). Let the bent lever ACB turn upon its centre c, by the action of the forces P and w applied obliquely to the arms CA and CB. Let CQ and CR be perpendiculars from the centre of motion c, upon the directions of the forces, and let the lever ACB be moved into the new position acb very near to the first. From a let fall ae perpen B W R a Fig. 69. A dicular to PA produced; then while the extremity A of the arm CA describes the arc Aa, the force P will have moved, in the line of its action, through Ae. Now when the change of position is indefinitely small, the circular arc Aa becomes a straight line perpendicular to CA, and ▲ e a is a right-angled triangle; moreover, if Aa be the space moved over by the extremity of the arm CA, the space moved over by P, in the direction Pe of its action, will be ae. Because CAа is a right angle, the Lane is equal to the ZACQ, and the triangles Aea and ACQ are equiangular, now by the equality of moments, Art. 117., we have PXCQ WXCR; multiplying these equations together PX space moved by P=Wx space moved by w, that is, work of P=work of w. Now as this equality will be true for any number of small motions that may be given in succession to the lever, provided only the forces are constantly in equilibrium, it follows that the equality will hold true for any definite motions that may be given. WHEEL AND AXLE. 125. This mechanical power is only another form of the lever, where the power is made to act without intermission; in its most simple form, it consists of a horizontal axle A and large wheel R, which turn upon two pivots supported in gudgeons. A cord wrapping round the axle A sustains the weight w, and another cord wrapping round the wheel R, in a contrary direction, sustains the power P. These forces always act in the direction of a tangent to the circle. Here the leverage of Fig. 70. the power is the radius of the wheel, and the leverage of the weight is the radius of the axle; hence we have 126. This figure represents a combination of wheels and axles. F is a wheel, to which the power Let R the radius of the wheel duced on the cord CA; then we have for the conditions of equilibrium PXR=QXR, and Q× R1=w× r1, and multiplying these equations together, PXRXQXR1 =Q ×? × W × 19 thus it appears that the power multiplied by the radii of the wheels is equal to the weight multiplied by the radii of the axles. 127. In the compound wheel and axle, represented in fig. 72. let R=the radius of the large axle a; r=the radius of the small axle D; and the length of the handle PO. Now as the weight w is suspended by the two cords CD and AB, they will each have a tension ofw, hence we have by the equality of moments B Fig. 72. W D which is the expression for the advantage gained. This evidently increases with the smallness of the difference of the radii of the axles. Cogged or Toothed Wheels. 128. Let D be a cogged wheel turning upon the same axis as the wheel c; Q another cogged wheel, acted upon by the former, be turned in the contrary direction, and thus the cord iw will be coiled up upon the axle 1, and the weight w will be raised. Let R the radius of the wheel, a; r=the radius of the toothed wheel D; R1=the radius of the toothed wheel Q; r1=the radius of the axle I; then proceeding as in Art. 126., we have for the conditions of equilibrium Let n the number of teeth in D, and n1=the number in Q; then since the number of teeth in the wheels are proportional to their The principle of the equality of work applied to the wheel and axle. 129. As an example, let us take the combination of wheels and axles described in Art. 128., see fig. 73. Let the axle I, with its toothed wheel Q, turn round once; then adopting the notation of Art. 128., we have No. turns of A with its cogged wheel D= = .. Space moved over by w=2πr1, No. teeth in Q No. teeth in D circum. Q_R1 circum. D r Now by multiplying each side of eq. (1), Art. 128., by 2π we get r |