Ex. 4. A common steelyard is 12 inches long, and with the scale-pan weighs 1 lb., the centre of gravity of the two being 2 inches from the end to which the scale-pan is attached; find the position of the fulcrum when the movable weight is 1 lb. and the greatest weight that can be ascertained by means of the steelyard is 12 lbs. Ans. 1 in. from scale-pan. Ex. 5. The movable weight of a common steelyard is 6 oz. A tradesman diminishes its weight by half an ounce: of how much is a person defrauded who buys what appears to weigh 6 lbs. by this steelyard? Ans. oz. Ex. 6. Find the length of a Danish steelyard, weighing 1 lb., when the distance between the graduations 4 lbs. and 5 lbs. is 1 inch. Ans. 30 in. 153. Roberval's Balance. This consists of four uniform rods, AB, BD, DC, CA, freely jointed at their extremities and forming a parallelogram. The rods AB, CD can turn about pivots at their middle points E, F, which are fixed in a vertical support. The rods AB, CD are similar in every respect, as are the rods AC, BD. Equal scale-pans are rigidly connected with AC and BD. The advantage of this balance is that it does not matter whereabouts the scale-pans the weights to be compared are placed. Let the weight P, when placed in the scale-pan attached to AC, counterbalance the weight Q placed in the other scale-pan. If now the B Fig.126 system be supposed displaced by the beams AB, CD turning through a small angle, it is clear that the centres of mass of AB, CD suffer no displacement, while that of BD and its scale-pan is raised or lowered through a vertical distance p, say, and the centre of mass of AC and its scale-pan is lowered or raised through the same distance. The virtual work done by the weight of BD will be equal to, but of opposite sign to, that done by the weight of AC. Also the algebraical sum of the virtual work done by the internal forces of the system is zero. The equation of virtual work is therefore Pp - Qp=0, since P, Q move through the same vertical distance as AC, and BD viz. p; therefore P=Q. This result holds wherever P and Q are placed in their respective scale-pans, i.e. whatever be their distances from the vertical support. 154. The Differential Wheel and Axle. In order to raise a very large weight by means of a comparatively small power, with the help of the ordinary 'wheel and axle', it would be necessary to make either the radius of the wheel inconveniently large, or else that of the axle so small that it would be unable to bear the strain put upon it. This difficulty is got over in the Differential Wheel and Axle'. This consists of two axles B and C, of different radii, rigidly connected together and turning about their common axis AE, which is horizontal and turns in fixed sockets. The power P is applied at right angles to the axis, and at the end of an arm AD, the 'wheel'; the weight W is attached to a pulley supported by a rope which is wrapped one way round B, and the other way round C: P and the rope round the thicker axle B tend to turn the machine in opposite directions. To find the Conditions of Equilibrium. Let a be the length of AD, b, c the radii of B, C respectively, and T the tension of the rope supporting the pulley. Since the pulley is in equilibrium 2T=W. Since the machine is in equilibrium, taking moments about the axis AE, we have or Pa Tb+Tc=0, ... Pa=T(b-c) = } W (b − c), P: W=b-c: 2a. Hence by making the radii of B and C as nearly equal as we please, the weight which a given power P can raise, may be increased to any extent. The principle of work also enables us to obtain this result very easily. In practice, this machine is useless, as in order to raise the weight through an appreciable height, the length of rope required would be very great. This difficulty is however got over in a modification of the differential wheel and axle, known from the name of the patentee, as Weston's Differential Pulley. In the Differential Pulley shewn in figure 128, an endless chain passes over a fixed pulley B, under a movable pulley to which the weight is attached, and then over another fixed pulley C, a little smaller than, but coaxial with B: the two ends of the chain are jointed so as to form a loop, the Power is applied to the right-hand portion of the loop: to prevent the chain from slipping, there are cavities formed in the circumferences of the upper pulleys into which the links of the chain fit, The condition of equilibrium is obtained as in the differential wheel and axle, and is the same, if we write b for a, i.e. is P: W b-c: 2b, where b is the radius of the larger fixed pulley, c that of the smaller. 155, Hunter's Differential Screw. This consists of a screw AD which works in a fixed nut CC'. AD is hollow and has a thread cut inside it, in which a solid screw DE works. DE is prevented from turning round by some means, for instance, by means of a rod FEF rigidly A W connected with it, and whose ends work in smooth grooves, so that the screw DE can only move in a direction parallel to its axis. The weight W is the resistance exerted by any substance placed between E and the base GG' of the framework CGG'C'. The power P is applied at the extremity of the arm AB which is at right angles to and rigidly connected with the screw AD. Condition of Equilibrium. Let a be the length of AB, h, l' the distances between consecutive threads of AD, DE respectively. Let us see the effect of the arm AB making a complete revolution. AD will clearly descend through a distance h: DE cannot turn with AD, and therefore will move upwards relatively to AD through a space h', i.e. will actually descend through a space h-h'; this is therefore the distance through which the weight is moved. Let us suppose the virtual displacement made to be that which would be produced by P moving its point of application through a small angle e, so that in consequence the weight descends through a distance x: as the distance through which DE descends is proportional to the angle through which AD turns, x/(h-h')=0|2π. As P and W are the only forces that do work during the above displacement, the equation of virtual work is P. a0-Wx=0, .. P. 2α=W (h – h'). This relation might have been obtained by an extension of the method adopted in Art. 146. It is clear that by making h and h' sufficiently nearly equal, we can make W/P as great as we please; whereas the same result is obtained in the simple screw only by making a inconveniently large, or by making h so small that the thread is too weak to support the pressure on it. EXAMPLES ON CHAPTER VII. 1. If a power P acting horizontally will support a weight W on a plane of inclination a, and would also support it on a plane of inclination ẞ, acting parallel to the plane, the pressure on the plane in the former case being double that in the latter, prove that a == cos1 (1). 2. If in the first system there be two pulleys, the fixed ends only of the strings being parallel, and the power horizontal, prove that the mechanical advantage is 3. 3. In the first system the weights of the pulleys beginning with the highest are in A. P. and a power P supports a weight W; the pulleys are then reversed, the highest being placed lowest and so on, and now W and P when interchanged are in equilibrium: shew that n (W+P) = 2W', where W' is the total weight of the pulleys and n is the number of them. 4. If there be n pulleys in the third system, and if the string which goes over the lowest have the end at which the power is usually hung, passed under another movable pulley, over a fixed pulley, and then attached to the weight W; and if the weight of each pulley be w and no other power be used, prove that IV (3.2′′-1 - n − 1)w. 5. In a weighing machine constructed on the principle of the common steelyard the pounds are read off by graduations reaching from |