LECTURE VII. CONTENTS.-The Wheel and Compound Axle, or Chinese Windlass-The Principle of Moments applied to the Wheel and Compound Axle-The Principle of Work applied to the Wheel and Compound Axie-Examples I. II.-Weston's Differential Pulley Block-The Principle of Work applied to Weston's Differential Pulley Block-Experiment I.— Cause of the Load not overhauling the Chain-Questions. The Wheel and Compound Axle, or Chinese Windlass. -This ingenious contrivance was first devised by the Chinese for the purpose of lifting weights. The theoretical mechanical advantage is very great, but it possesses the disadvantage of requiring a long length of rope to lift the weight a small height. Its construction and action will be easily understood from the accompanying side and end views, which are taken from a model SIDE VIEW. W END VIEW W (Without End Bearing). THE WHEEL AND COMPOUND AXLE. THE WHEEL AND COMPOUND AXLE. 57 made in the author's engineering workshop for the purpose of demonstrating its action and efficiency to his students. The Principle of Moments applied to the Wheel and Compound Axle.-Taking moments about the axle, we have, when there is equilibrium between P and W, The Principle of Work applied to the Wheel and Compound Axle.-Neglecting friction, and supposing the rope to be perfectly flexible, cause the wheel to make one complete revolution in the direction shown by the arrow near its circumference on the end view. Then, by the principle of work, Or, i.e., Or, P × circumference = W × of the difference of the cir of wheel cumferences of the larger and smaller axles.* P × 2πR = W × 1(2πr ̧ — 2π1o1⁄2) (Dividing both sides of the equation by 27)— Which is the same result as the one above; consequently the principle of moments and the principle of work agree. EXAMPLE I. In a compound wheel and axle, where the weight hangs on a single movable pulley, the diameters of the two portions of the axle are 3 and 2 inches respectively, and the lever handle which rotates the axle is 12 inches in length. If a force W * If is raised the circumference of the larger circle on one side, W 2 then is lowered at the same time on the other side, the circumference 2 of the smaller axle; consequently W will be elevated a distance equal to half the difference of the circumferences of two axles, or= (2πr1 — 2πr1⁄2). of 10 lbs. be applied to the end of the lever handle, what weight can be raised? ANSWER.--Here P = 10 lbs. ; R = 12" ; r1 = 1.5" and "= 1′′. IO X I2 = - W × 1(1.5 − 1) = W × 1 × 1 = {W EXAMPLE II. In a compound wheel and axle, let the diameter of the large axle be 6 inches, and that of the smaller axle 4 inches, and the length of the handle 20 inches; find the ratio of the velocity of the handle to that of the weight raised. ANSWER.-Here R = 20"; "1 = 3" ; ï1⁄2=2′′. By the principle of moments and work But by the principle of work Px its distance = W x its distance IX P's distance = 40 × W's distance Weston's Differential Pulley Block. This practical application of the Chinese windlass is simply a compound axle without the wheel. Or, where R=r1. where R is the radius of the larger axle or pulley, and the radius of the smaller one. After describing Weston's differential pulley block, we will deduce this formula from the "principle of work" by the same kind of reasoning as we adopted in the case of the wheel and compound axle. We leave the student, however, to apply the "principle of moments," whereby he should get the same results. WESTON'S DIFFERENTIAL PULLEY BLOCK. 59 As will be gathered from an inspection of the accompanying outside view and the small diagram showing the directions of the forces and their arms, it will be seen that the apparatus consists of three parts (1) an upper block; (2) an endless chain; (3) a movable lower block or snatchblock. The upper block has a hook with swivel joint, from which the iron frame is suspended. In the centre of this frame is a turned steel axle on WESTON'S DIFFERENTIAL thereby prevent it slipping over the surface of the pulleys. The lower or movable pulley is simply an ordinary smooth V-grooved pulley with swivel and hook like that already described under the heading "Snatch Block." The endless chain is an ordinary openlinked chain of uniform pitch and size of link. It passes from the position where the hand or pull, P, is applied, over the larger pulley of the upper block, underneath the lower pulley, over the smaller of the upper block pulleys, and back to the starting-point. (See also the small figure.) When a pull, P, is applied at this part of the chain (if there were no friction), it would be transmitted with undiminished value throughout its whole length where the tension can act; but, as we shall see afterwards, a large proportion of this force is absorbed in overcoming friction. The stress due to the load W is divided equally between the two vertical parts of the chain connected to the lower block, and if W is moved through any W 2 distance, the stress must act through double that distance. The Principle of Work applied to Weston's Differential Pulley Block and Tackle.-Theoretically (i.e., leaving friction out of account, the weight of the hanging part of the chain and the weight of the lower block), we have by the principle of work, in one revolution of the upper pulleys Px its distance = W x its distance. Px circumference of the larger pulley P x 2πR = 2 (2πR-2πr) (1) The Theoretical Mechanical Advantage or ratio of W to P is found directly from the above equation by simple transposition. (2) The Velocity Ratio (or ratio of the distance passed through by P to the distance passed through by W in the same time) is also found in the same way. Or, the velocity ratio has the same numerical value as the theoretical advantage. EXPERIMENT I.-With a Weston's differential pulley block, having in the upper block one pulley with an effective radius of 4'" (i.e., from the centre of the pulley to the centre of the chain which passes round it), and a smaller pulley with an effective radius of 31", you can just lift a total load of 100 lbs. (including the dead weight, the lower block, and the hanging parts of the chain) by a pull of 20 lbs. on the chain. * Dividing numerator and denominator by π does not alter the fraction. |