his sustained by one fold of the cord and w by 12 folds: P will equipoise 1200 (for 12 x 100=1200) and a small additional force will raise w higher. THE INCLINED PLANE. An inclined plane makes an acute or an obtuse angle with the plane of the horizon. The force with which a body descends down an inclined plane, is to that with which it would descend freely as the elevation of the plane is to its length. b e 1200 C e Let a b, be an inclined plane, e b, its elevation, and a c, a plane parallel to the horizon, the body e will remain at rest on any part of the horizontal plane, or a small impulse will roll it in any direction. But let the body e be placed on the inclined plane a b, the force with which it will descend along the plane, will be to its absolute gravity, as the elevation of the plane to its length reciprocally. EXAMPLE. A man who exerts a force of 100 lb. in rolling a body up an inclined plane of the same proportions, will sustain 5 times as much, and a small additional force will roll the body up the plane. The motion of a body descending down an inclined plane, is uniformly accelerated; therefore whatever was demonstrated concerning the gravitation of bodies, is equally applicable to their descent down inclined planes; the motion in both cases being the effect of the same power: Therefore, The velocity acquired in any given time, by a body descending down an inclined plane, is to the velocity acquired by a body falling freely in the same time, as the elevation of the plane is to the length; and The time in which a body moves down an inclined plane, is to that in which it would fall perpendicularly from the same height, as the length of the plane is to its elevation. THE SCREW. The screw is a cylinder which has either a prominent part or a hollow line passing round it in a spiral form, so inserted in one of the opposite kind that it may be raised or depressed at pleasure, with the weight upon its upper part or suspended beneath its lower surface. In the screw, the equilibrium will be produced when the power is to the weight as the distance between the contiguous threads, in a direction parallel to the axis of the screw, is to the circumference of the circle described by the power in one revolution. The screw is not a simple machine, because it requires a lever or winch to work it, and then it becomes a compound machine of very great force. Cut a piece of paper to the figure of the inclined plane, a b y, and fold it round a pencil, it will then convey a perfect idea of the screw. While the screw performs one revolution, the weight may be supposed to be raised up one of the spirals or an inclined plane, whose height is equal to the distance α b between two contiguous spirals, and whose base, y x, the circumference of the cylinder and length; x o, is the spiral line. THE WEDGE. The wedge is composed of two inclined planes, whose bases are joined: Its properties are so well known that it is superfluous to say more than that it is a power of very great force and use. When the resisting forces, and the powers which act on the wedge, are in equilibrio, the weight will be to the power, as the height of the wedge to a line drawn from the middle of the base to one side, and parallel to the direction in which the resisting force acts on that side. SCHOLUM. In theory of the mechanical powers, we suppose all planes and bodies perfectly smooth; levers to have no weight; cords to be perfectly pliable, and the parts of machines, to have no friction; but it is found, in practice, that few compound machines but require a third part more to work them when loaded, than what is sufficient to constitute an equilibrium between the weight and the power. The inequality of the surface, on which any body moves, is called friction, which prevents the accurate agreement of many experiments in mechanics with theory. Friction is an uniformly retarding force in hard bodies, not subject to alteration by velocity, except when the body is covered with a woollen cloth; in that case, the friction ceases a little with the velocity. Friction increases in a less ratio than the weight of the body; being different in different bodies. The smallest surface has the least friction. Wheel carriages are used to avoid friction as much as possible: A wheel turns round on its axis, because the several points of its circumference are retarded in succession by attrition, while the opposite points move freely. Large wheels meet with less resistance than smaller from external obstacles, and from the friction of the axle, and are more easily drawn, having their axles level with the horses; but in uneven roads small wheels are used, that in ascents the action of the horse may be nearly parallel with the plane of the ascent, and therefore may have the greatest effect. Vibration of Pendulums. A pendulum is a heavy body hanging by a cord or wire, and movable with it upon a centre. The vibrations of a pendulum are produced by the force of gravitation. The time of ascent and descent of a pendulum, is equal to the time in which a body falling freely, would descend through eight times the length of the pendulum. The times in which pendulums of different lengths perform their vibrations, are as the square roots of their lengths. The lengths of pendulums, are to each other, reciprocally, as the square of their vibrations in the same space of time. A pendulum, whose length is 39.2 inches from the point of suspension to the centre of oscillation, will make 60 vibrations in a minute. This is called standard length. EXAMPLE. What difference is there in the length of two pendulums, the one vibrating seconds, or 120 times in a minute, the other vibrating double seconds, or 30 times in a minute? 120,2=14400 square vibrations. 30,2 900 do. 3600 do. do. in 1 hour. Then it will be reciprocally, 60,2 If the length of a pendulum be given, and the number of vibra tions in a minute required. RULE. As given length: standard length :: square of 60 vib. : square of the number required. EXAMPLE. Required the number of vibrations in a minute, by a pendulum 156.8 inches long. 6023600 in. in. sq. vib. 89.vib. Then as 156.8.: 39.2 :: 3600 : 900 and/900 30 vibrations, Ans. What difference is there between the number of vibrations made by a pendulum 6 inches long and another of 12 inches, in an hour's time? Ans. 2695.08. What difference is there in the length of two pendulums, the first swinging 30 and the second 100 times in an hour? Ans. 42806 feet 4.8 inches. Required the length of the pendulums that will swing once in a third, once in a minute, once in an hour, and once in a day. 1. A pendulum that vibrates thirds .0108 inches. Ans. 2. Once in a minute, 2 miles 1 fur. 32 poles 4 yards. Miscellaneous Exercise. English 1. The globe of the earth, under the line, is 360 degrees in circumference, each degree 69 miles; and this body being turned on its own axis in 23 hours 56 minutes, at what rate an hour are the inhabitants of Bencoolen, situate under the line, carried about from west to east by this rotation? Ans. 104545 miles. 2. Three merchants, A, B and C, trade in company; at making up accounts, it appears that A and B together gained £13 10; B and C together £12 12; and A and C together £11 16 6: What did they severally gain? Ans. A £6 7 3, B £7 2 9, and C £5 9 3. 3. Some others advance in trade, viz. W, X and Y raised £350 10; W, X and Z £344 10; X, Y and Z made up a stock of £400, and W, Y and Z contribute £378 4: in the conclusion, they parted with their joint property for 450 guineas: What did they gain or lose by their adventure? Ans. £18 11 4 sterling loss. 4. There is a mast or pole, of its length stands in the ground, 12 feet of it in the water, and of its length in the air, or above water: What is the whole length? Ans. 216 feet. 5. What quantity of water must I add to a pipe of mountain wine, value £33, to reduce the first cost to 4s. 6d. per gallon? Ans. 20 gallons. 6. There are 2 pieces of linen; the one is 9 yards shorter than the other, and cost £3 18.; the other piece, at the same price, cost £3 128. I demand how many yards are in both pieces, and the price of 1 yard. Ans. longest 60, shortest 51 yards, at 1 shillings per yd. 7. A piece of satin cost a certain sum, and being sold for £3 10s. there is lost in a shilling: What was the first cost? Ans. £5 16 8. 8. With 13 gallons Canary, at 6s. 8d. the gallon, I mingled 20 gallons white wine, at 5s. and 10 gallons cyder at, 3s. the gallon: How am I to sell a quart of this mixture, so as to clear 10 per cent.? Ans. 1627d. per quart. 9. A is dispatched on a commission from Washington to Boston, distant by computation, say 350 miles, and his route is settled 22 miles a day; 4 days after, B is sent after him with fresh orders and is to travel 32 miles a day: Whereabout on the road will B overtake A? Ans. 683 miles on this side Boston. 10. If 6 lb. pepper be worth 13 lb. ginger, and 19 lb. ginger be worth 44 lb. cloves, and 10 lb. cloves be equal to 63 lb. of sugar, at 5d. per lb. what is the value of 1 cwt. of pepper? Ans. £7 19 3. 11. If 30 men can perform a piece of work in 11 days, how many men must be added to finish another 4 times as big in of the time? Ans. 600 men. 12. A may-pole, 50 feet 11 inches high, at a certain time of day casts a shadow 98 feet 6 inches long: I would hereby find the breadth of a river that is running 20 feet 6 inches from the foot of a steeple, 300 feet 8 inches high, when the extremity of the shadow of the steeple reaches 30 feet 9 inches beyond the Ans. 530 feet 5 inches, nearly. stream. 13. A tradesman increased his estate annually, abating £100, which he usually spent in his family; and at the end of 34 years found that his neat estate amounted to £3154 11 8: What had he at out-setting? Ans. £1411 12 93. 14. A can do a piece of work in 10 days, B alone in 13 days: Set them both about it together, in what time will it be finished? Ans. 5 days. 15. A cistern is supplied with water by one pipe, of such bigness that if the cock A, at the end of the pipe, be set open, the cistern will be filled in an hour; but at the bottom of the cistern are two other cocks, B and C, whose capacities are such that, by the cock B, set open alone, (all the rest being stopt) the cistern, supposed to be full, will be emptied in 14 hours, and by C alone in 2 hours. Now, because more water will be infused by the cock A, than can be expelled by the cocks B and C in the same time, the question is, in what time the cistern will be filled, if all the three cocks are set open at once? Ans. 1h. nearly. |