A; the teeth, or cogs, of this wheel work into the spaces of the toothed wheel B, and the resistance is attached to a rope wound round the arbor p Fig. 85. R teeth in one wheel should be large enough to receive the teeth of the other wheel, but not large enough to allow a great deal of play. The teeth should always come in contact at the same distances from the centres of the wheels, and those distances are taken as the radii of the wheels themselves. Denote the power by P, the resistance by R, the crank-arm by c, the radius of the wheel A by r, that of the wheel B by r', that of the arbor by r", and suppose the power and resistance to be in equilibrium; then will the pressure due to the action of the power tend to turn the wheels in the direction of the arrow heads. This tendency will be counteracted by the pressure of the resistance tending to produce motion in a contrary direction. If we denote the pressure at the point C by R', we should have, from what has preceded, Pc R'r and R'r' Rr"; That is, the power is to the resistance as the continued product of the alternate arms of lever, beginning at the resistance, is to the continued product of the alternate arms of lever beginning at the power. Had there been any number of wheels in the train lying between the power and resistance, we should have found similar conditions of equilibrium. EXAMPLES. 1. A power of 5 lbs., acting at the circumference of a wheel whose radius is 5 feet, supports a resistance of 10 lbs. applied at the circumference of the axle. What is the radius of the axle ? Ans. 14 inches. 2. The radius of the axle of a windlass is 3 inches, and the crank-arm 15 inches. What power must be applied to the crank-handle, to support a resistance of 180 lbs., applied to the circumference of the axle ? Ans. 36 lbs. 3. A power P, acts upon a rope 2 inches in diameter, passing over a wheel whose radius is 3 feet, and supports a resistance of 320 lbs., applied by a rope of the same diameter, passing over an axle whose radius is 4 inches. What is the value of P, when the thickness of the rope is taken into account. Ans. 43 lbs. The Screw. Fig. 86. 97. The screw is essentially a combination of two inclined planes. It consists of a solid cylinder, called the cylinder of the screw, which is enveloped by a spiral projection called the thread. The thread may be generated as follows: let an isosceles triangle be placed so that its base shall coincide with an element of the cylinder of the screw, and so that its plane shall pass through the axis. Let the triangle be revolved uniformly about the axis, and at the same time be moved uniformly in the direction of the axis, at such a rate that it shall pass over a distance in this direction equal to the base of the triangle during one revolution. The solid generated by the triangle is the thread of the screw. The two sides of the triangle generate helicoidal surfaces, which constitute the upper and lower surfaces of the thread. Every point in these lines generates a curve called a helix, which is entirely similar to an inclined plane bent around a cylinder. The vertex generates what is called the outer helix, and the two angular points of the base trace out the same curve, which is the inner helix. The screw just described is called a screw with a triangular thread. Had we used a rectangle, instead of a triangle, and imposed the condition, that the motion in the direction of the axis during one revolution, should be equal to twice the base, we should have had a screw with a rectangular thread, as in the figure. The screw works into a piece called a nut, which is generated in a manner entirely analogous to that just described, except that what is solid in the screw is wanting in the nut; it is, therefore, exactly adapted to receive the thread of the screw. Sometimes, the screw remains fast, and the nut is turned upon it; in which case, the nut has a motion of revolution, combined with a longitudinal motion. Sometimes, the nut remains fast, and the screw is turned within it, in which case, the screw receives a motion in the direction of its axis, in connection with a motion of rotation. The conditions of equilibrium are the same for each. In both cases, the power is applied at the extremity of a lever; when in motion, the point of application describes an ascending or descending spiral, resulting from a combination of the rotary and the longitudinal motion We shall suppose the nut to remain fast, and the screw to be movable, and that the resistance acts parallel to the axis of the screw. If the axis is vertical, and the resistance a weight, we may regard that weight as resting upon one of the helices, and sustained in equilibrium by a force applied horizontally. If we suppose the supporting helix to be developed on a vertical plane, it will form an inclined plane, whose base is the circumference of the base of the cylinder on which it lies, and whose altitude is the distance between the threads of the screw. Let AB represent the development of this helix on a vertical plane, and denote by F the force applied parallel to the base, and immediately to the weight R, to sustain it on the plane. We shall have (Art 85), F: R:: BC: AC. Combining this proportion with the preceding one, and recollecting that AC 2r, we deduce the proportion, = That is, the power is to the resistance as the distance between the threads is to the circumference described by the point of application of the power. By suitably diminishing the distance between the threads, other things being equal, any amount of mechanical advantage may be obtained. The screw is used for producing great pressures through very small distances, as in pressing books for the binder, packing merchandise, expressing oils, and the like. On account of the great amount of friction, and other hurtful resistances developed, the modulus of the machine is very small. The Differential Screw. 98. The differential screw consists essentially of an ordinary screw, as just described, into the end of which works a smaller screw, having its axis coincident with the first, but having its thread turned in a contrary direction; that is, it is what is technically called a left-handed screw, the first screw being a right-handed one. The distance between the threads of the second screw is somewhat less than that between the threads of the first screw, and this difference may be made as small as desirable. The second screw is so arranged that it admits of a longitudinal motion, but not of a motion of rotation. By the action of the differential screw, the weight is raised vertically through a distance equal to the difference of the distances between the threads on the two screws, for each revolution of the point of application of the power. For, were the first screw alone to turn, the weight would be raised through a distance equal to the distance between its threads; but, because the second screw is a left-handed one, this distance will be diminished by a distance equal to that between its threads. We may, therefore, write the following rule: The power is to the resistance as the difference of the distances between the threads of the two screws is to the circumference described by the point of application of the power. Endless Screw. When 99. The endless screw is a screw secured by shoulders, so that it cannot be moved longitudinally, and working into a toothed wheel. The distance between the teeth should be nearly equal to the distance between the threads of the screw. the screw is turned, it imparts a rotary motion to the wheel, which may be utilized by any mechanical device. The conditions of equilibrium are the same as for the screw, the resistance in this case being offered by the wheel, in the direction of its circumference. Fig. 88. Machines of this kind are used in determining the number of revolutions of an axis. An endless screw is arranged to turn as many times as the axis, and being connected with a train of light wheel-work, the last piece of which bears an index, the number of revolutions can readily be ascertained |