CASE III. 3. What weight will be balanced by a power of 100 lbs. attached to a cord, that passes over 2 moveable pulleys? Ans. 400 lbs. CASE IV. 4. If a cord that passes over two moveable pulleys be attached to an axle 6 inches in diameter, and if the wheel be 60 inches in diameter, what weight may be raised by the pulley, by applying 100 pounds to the wheel? Ans. 4000 lbs. INCLINED PLANE. An inclined plane, is a plane which makes an acute angle with the horizon. The motion of a body descending down an inclined plane is uniformly accelerated. The force with which a body descends by the force of attraction down an inclined plane is to that, with which it would descend freely as the elevation of the plane is to its length, or as the size of its angle of inclination to radius. To find the power that will draw a weight of an inclined plane. Rule.-Multiply the weight by the perpendicular height of the plane, and divide the product by the length. 1. An inclined plane is 60 feet in length, and 15 feet perpendicular height, what power is sufficient to draw up a weight of 1000 lbs. Ans. 250 lbs. 2. A certain Railroad, 1 mile in length has a perpendicular elevation of 20 feet, what power is sufficient to draw a train of baggage cars, weighing 79200 lbs. up this elevation? Ans. 300 lbs. 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 up per 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 two contiguous threads, in a direction parallel to the axis of the screw to the circumference of the circle described by the power in one revolution. To find the power that should be applied to raise a given weight. Rule.-As the distance between the threads of the screw is to the circumference of the circle described by the power, so is the power to the weight to be raised. Note.-One-third of the power is lost in overcoming friction. 1. If the threads of a screw be 1 inch apart, and a power of 100 lbs. be applied to the end of a lever 10 feet long, what force will be exerted at the end of the screw? Ans. 75398.20 + lbs. С The wedge is composed of two inclined planes whose bases are joined. When the resisting forces and the power which acts on the wedge are in equili- F. E brio, 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. To find the force of the Wedge. RULE.-As the breadth or thickness of the head of the wedge is to one of its slanting sides, so is the power which acts against its head to the force produced at its side. 1. Suppose 100 pounds to be applied to the head of a wedge that was 2 inches broad, and 20 inches long, what force would be effected on each side? Ans. 1000 lbs. THE WEDGE. CASE I. ON THE STEAM ENGINE. Steam at the temperature of 212° is 1800 times its bulk in water; or, one cubic foot of steam, when its elasticity is equal to thirty inches of mercury, contains one cubic inch of water. Therefore, when an engine in good order, is performing its regular work, the effective pressure may be taken at eight pounds on each square inch of the surface of the piston. To calculate the power of an Engine. It has been demonstrated in the Franklin Institute, of Philadelphia, that a horse can draw 200 lbs. at the rate of 21 miles an hour, or 220 feet in a minute with a continuance drawing over a pully, that is, 200 x 220 = 44000 lbs. at 1 foot per minute, or 1 lb. at 44000 feet per minute. Rule 1.--Multiply the area of the cylinder by the effective pressure, say 8 lbs., the product is the weight the engine can raise. Multiply this weight by the number of feet the piston travels in one minute, the product will give the momentum, divide this momentum by a horse power, and the quotient will be the number of horse power in the engine. 2. The velocity of an engine being 220 feet per minute, 25 inches of the area of the cylinder is equal to one 1. What is the power of an engine, the cylinder being 42 inches diameter and stroke 5 feet? 422 x .7854 x 8 x 210 74.44 horse power. 44000 2. What size of cylinder will a 60 horse power engine require when the stroke is 6 feet? 44000 x 60 = 1447.39 + area of cylinder. 228 x 8 Examples calculated by Rule Second. 1. What diameter is the cylinder of a 40 horse engine, common pressure? ✓ 40 X 25 35.7 say 354 inches diameter. .7854 CASE II. To find the power to lift a weight at any velocity. Rule.-Multiply the weight in pounds, by the velocity in feet, and divide by the horse power, the quotient will be the number of horse power. TABLE SECOND. 66 66 When the effective pressure | The arca equal to one 3.70 inches. 4.17 43 4.65 38 5.26 33 6.06 28 7.14 23 8.70 11.11 13 15.46 8 25.00 18 66 66 1. What diameter is the cylinder of a 40 horse engine, effective pressure 33 lbs. on the square inch? ✓ 40 x 6.06 = say 17 inches diameter. 2. The cylinder of an engine is 40 inches diameter, and the effective pressure is 20 lbs. on the square inch, what is the power of the engine? 402 x .7854 = 1256.64 = 157.08 horse power. GENERAL THEOREMS.* THEOREM 1. When the sum and difference of any two numbers are given, to find the numbers. Rule.—To half the sum add half the difference, for the greater number. From half the sum, take half the difference for the lesser number. THEOREM 2. The product of the sum and difference of any two numbers is equal to the difference of their squares. THEOREM 3. If the difference of the squares of two numbers be divided by their difference, the quotient will be the sum; and if by the sum of the numbers, the quotient will be the difference. THEOREM 4. If a number be divided into any two parts, the square of the number is equal to the sum of the squares of the two parts, and twice the product of those parts. THEOREM 5. If the difference of the cubes of any two numbers be divided by their difference, the quotient arising will be equal to the sum of the squares of the two numbers together with their product. THEOREM 6. When the sum and product of two numbers are given to find the numbers * These Theorems should be committed to memory. |