Also, by taking moments about F, we have Wq+r sin λ. (P2 + W2+2PW cos )...(1). If P and W are in the same direction this becomes Pp-r sin x) = W (q+r sin λ). If P could only just balance W, or, in other words, were W on the point of moving P, the relation would be Pp=Wq-r sin λ. √(P2 + W2+2PW cos 0). 138. To find the efficiency (E) of the rough Lever. Let P be the power just required to move W, when the fulcrum is perfectly smooth. cos Ꮎ :0); . . pq (1 − E) = r sin λ . √√(q2 + p2E2 + 2pq E cos 0), which gives us E. If P and W are in the same direction E becomes q (p −r sin λ)/{p (q+r sin λ)}. 139. The Wheel and Axle. This machine consists of a cylinder a (the wheel) with a groove cut round the circumference, and a cylinder b of smaller radius (the axle). The two form one rigid body and have a common horizontal axis cc', at the ends of which are two pivots c and c', resting in fixed sockets so that the whole can turn about this axis. The power P is applied tangentially at the circumference of the wheel, generally by means of a rope, while α W Fig. 112, the weight is suspended by a rope which is wound round the axle so that it tends to turn the machine in the opposite direction to the power. The apparatus for drawing a bucket of water out of a well is frequently a machine of this kind, the power being applied by means of a handle attached to the wheel instead of by a rope. A windlass for hauling up an anchor on board ship is a modification of the wheel and axle, in which the common axis is vertical, and the power is applied at the end of poles which project from the wheel so as to form radii produced. Condition of Equilibrium. The Wheel and Axle, as before stated, is a kind of lever, and we can shew as in Art. 74 that the condition of equilibrium is that the moment of P about the axis should be equal and opposite to the moment of W, i.e. that Px the radius of the wheel = W × radius of the axle. If the ropes be of considerable thickness, the tension of each may be supposed to act along its axis or central line, so that the condition of equilibrium becomes Rigidity of Ropes. We have hitherto supposed, that the ropes are perfectly flexible, i.e. that they offer no resistance to being bent. As a matter of fact when a rope is wound on to a drum, pulley or axle, it does offer a resistance, though none is offered when it is wound off. From experiments made on new dry ropes and tarred ones Coulomb has deduced the following empirical results. If a rope whose tension is T, is on the point of being wound on to a drum, the effect of the rigidity of the rope is the same as would be produced by increasing T by a certain amount T'. This amount T' is most simply expressed by the formula where a and b are constants depending on the rope, and R' is the effective radius of the drum, i.e. its actual radius + half the thickness of the rope. What is the precise meaning of the above statement? The force T exerted by the rope cannot be greater than itself; how can the rigidity of the rope increase the effect of T? As it cannot alter the magnitude and direction of T, it clearly can only increase T's effect by causing it to act at a greater distance from the drum than would be the case, if the flexibility were perfect. This is precisely what we should be led to expect from à priori considerations. Where the rope is being wound on, those fibres furthest from the drum will be stretched more than those nearer, and will therefore exert the greatest tensions-hence the resultant tension will act further from the drum than the central axis of the rope. At the point where the rope is being wound off, there seems no reason why one fibre should exert a greater tension than another so that the resultant tension will act along the central axis. Where must T act in order to have as great a moment about the axis a+bT of the axle or drum, as T+ acting along the central axis of the R rope? If the distance of the line of application of T from the central axis x, we must have be r clearly cannot be greater than the thickness of the rope. By using the method of Art. 137, the following results may be obtained. If the Power P applied to a wheel and axle of weight w, by means of a rope of thickness t1, be about to raise a weight W which is suspended by a rope of thickness t2, the relation between P and W is P (R+t-p sin λ) = W (r+t2+p sin λ)+wp sin λ, where R is the radius of the wheel, r that of the axle, p that of the pivots about which the whole turns and λ the angle of friction between the pivots and their bearings. If the rigidity of the rope to which W is attached be taken into account, the relation becomes where a and b are the constants mentioned above. In both these cases, P is applied vertically downwards. Ex. 1. Four sailors, each exerting a force of 112 lbs., can just raise an anchor by means of a capstan whose radius is 1 ft. 2 in. and whose spokes are 8 ft. long (measured from the axis). Find the weight of the anchor. Ans. 11 tons. Ex. 2. If the length of each of a pair of sculls be 8 ft. 6 in., and the distance from the hand to the rowlock be 2 ft. 3 in., find the resultant force on the boat when the sculler pulls each scull with a force of 25 lbs., assuming that the blade does not move through the water. Ans. 18 lbs. Ex. 3. A fly-wheel 10 ft. in radius weighs 15 tons, its axle is 6 in. in radius and revolves in bearings between which and it the coefficient of friction is 2: find the smallest weight which, hung from a band round the circumference of the wheel, will just turn it. Ans. 333 lbs. nearly. Ex. 4. Find the efficiency of a 'wheel and axle,' which weighs 50 lbs. and turns on pivots of in. radius, and coefficient of friction 1, when the Power acts vertically downwards, the radii of the wheel and axle are 2 ft. 8 in. and 5 in. respectively, and the weight to be raised is 500 lbs. Ans. 987. Ex. 5. Find the efficiency of the machine described in Ex. 4, everything remaining the same, except that the thickness of the ropes and their rigidity are to be taken into account; the ropes are 8 in. thick, and the constants a, b (Art. 139), are 8.6 and 18 respectively, provided that, in using the empirical formula of Art. 139, R' is expressed in inches, and T in lbs. Ans. 952. 140. The Pulley. A pulley-block consists of two plates or sheaves connected by an axle about which a circular disc, with a groove cut in its circumference, can turn. Rigidly connected with the axle is a hook to which a string can be attached so as to support the pulley, or by means of which the pulley can support a weight. Sometimes there are several discs, either turning about the same axle or placed one below another; they then form double, treble, &c. blocks. A rope passes along the groove in the circumference of the disc, and, as the latter is supposed smooth, the tension of the rope will be the same on both sides the pulley. When the block is fixed, the pulley is said to be fixed; otherwise it is called a movable pulley. If a fixed pulley be used to enable us to overcome resistance, the only object gained by the use of the pulley is that the force applied is enabled to counteract a force in a different direction, though not of greater magnitude. Fig.113. W When a single movable pulley is used, the weight W is attached to the block, and the power P is applied at one end of a rope which passes under the disc of the |