CHAPTER VII. MACHINES. 131. Ir is frequently desirable that we should be able to counteract one force by another, differing from it in magnitude, point of application, or direction, or in all three. To enable us to do this we employ machines more or less complicated. In Statics we suppose the machine to be in equilibrium under the action of the forces due to the geometrical conditions of constraint, the force at our disposal generally called the Power, and the force which we wish to counteract, generally called the Resistance or the Weight. It is found practically that, when the power is just on the point of overcoming the weight, other resistances are called into play, owing chiefly to the friction between the different parts of the machine, and the imperfect flexibility of ropes: all these resistances oppose the power, so that the latter has to be greater than would be necessary, were the machine a perfect one. If the weight were on the point of overcoming the power, these resistances would assist the latter. It is usual to call the resistance or weight, which it is the object of the machine to enable us to overcome, the useful resistance, while the other resistances are called wasteful resistances. When we take these latter into consideration, we shall suppose that motion is just about to take place, and that the power is overcoming the useful resistance. If motion just occurs, the work done by the power will equal that done against both the useful resistance and the wasteful ones; the former part of the work is termed useful and the latter lost work. 132. Def. When motion just takes place in a machine, the ratio of the useful work done to the whole work done in the same indefinitely short time is called the Efficiency of the machine. It is of course desirable to have the efficiency as near unity as possible. Let P denote the power, W the useful resistance, and W' the wasteful resistance. If P move its point of application through a small distance s, and in consequence the work done against W be w, and that done against W' be w', we have from the principle of virtual work, Ps=w+w the efficiency then is w/(w+w'). 0 Let P be the force which would just move W were there no wasteful resistance, then Ps=w by the principle of virtual work. Hence the efficiency = Ps/Ps=P/P, or the efficiency is the ratio of the power, which would just move the weight were there no wasteful resistance, to the actual power required. Unless otherwise stated, we shall suppose the machines perfect ones, i.e. with efficiency equal to unity, 133. The simple machines are, the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Screw and the Wedge. The principle of the wheel and axle is the same as that of the lever, and the screw and wedge are identical in principle with the inclined plane. 134. The Lever. This is a rigid rod, straight or curved, and free to turn about a fixed axis, which is called the fulcrum. The two parts into which the rod is divided by the fulcrum are called arms. Levers are usually classified as follows. In the lever of the first class, the fulcrum is between the power and the weight: a poker where the bar of the grate is used as the fulcrum, and a pair of scissors are instances of it. In the second class, the weight is between the fulcrum and the power, as in a wheelbarrow, where the point of the wheel in contact with the ground is the fulcrum, or in an oar, where the blade in contact with the water is the fulcrum, and the resistance is applied at the rowlock. In levers of the third class, of which a pair of shears and the human arm are examples, the power is between the fulcrum and the weight. In 135. The condition of equilibrium of a Lever. As in Art. 74 we can shew that the necessary and sufficient condition of equilibrium of any body whatsoever, which is free to turn about a fixed axis, and under the action of any number of forces, is, that the algebraical sum of the moments of the forces about the fixed axis be zero. the case of the simplest form of the lever the forces are generally only two, the power and the weight, acting in one plane, so that the condition of equilibrium becomes that the moment of P about the fulcrum should be numerically equal but of opposite sign to that of W. This condition may also be easily found by the Principle of Virtual Work. 136. To determine the pressure on the fulcrum wher the Lever is in equilibrium. Since the action of the fulcrum together with the power and the weight keeps the lever in equilibrium, the reaction on the fulcrum is obviously the resultant of the power and the weight. If, however, the lines of action of P and W are not in one plane, they do not reduce to a single resultant, and the pressure on the fulcrum is not a single force. We shall assume that the lines of action of P and W are in one plane. When the power and the weight are parallel, the reaction (R) of the fulcrum (F) is parallel to each of them; and in a lever of the first class, R=P+W, of the second class, R= W- P, of the third class, RP W. When the lines of action of the power and weight are not parallel but meet in C, let A, B be their respective R ΦΕ A Fig.110. B points of application, a, ẞ the angles, which their lines of action make with AB. It is required to find the magnitude of R, and the angle (0) its direction makes with AB. Since the lever is in equilibrium under the action of the three forces, R's line of action passes through C. = Also (Art. 18) sin ACF sin BCF W: P; Also : .. sin (0 − a) : sin (π – 0 – ß) = W : P ; cos é sin a + cos e sin ß W sin ẞ+P sin a P cos a W cos B R2 = P2 + Q2 + 2PQ cos ACB; 137. To find the relation between the Power and the Weight in a rough Lever, when the Power is on the point of moving the Weight. Let A, B be the points of application of the power (P) and the weight (W) respectively: let their lines of action A W Fig.III meet in C at an angle 0. The fulcrum is a rough solid cylinder, which passes through a cylindrical hole in the lever, just so much bigger in diameter that there is contact along one generating line only. Let the plane ABC, which we assume to be perpendicular to the axis of either cylinder, cut the hole in the circle DEG of radius r and centre F, D being the point where the line of contact meets the circle. Join DC, then the reaction of the fulcrum (R) acts along CD. Also, since the lever is on the point of turning round Fin the direction in which P tends to turn it, the reaction R will make with the normal FD an angle equal to λ, the angle of friction, and on the side which enables it to assist W. Let p, q be the perpendiculars from F on the lines of action of P, W respectively. Since R counteracts P and W, R2= P2+ W2+2PW cos 0. |