100 lbs. per ton. How many foot-pounds of work are expended in drawing 10 tons over 100 yards? Work Done against a Variable Resistance.-If the resist WORK VARYING ance varies whilst the force overcoming it acts through a known distance, then the work done will be measured by the product of the average resistance and the distance. If the resistance varies uniformly, its average can be found by adding its values at the commencement and end of the motion, and dividing by two. EXAMPLE V.-Explain the method of estimating the work done by a force, and define the unit of work. The surface of the water in a well is at a depth of 20 feet, and when 500 gallons have been pumped out, the surface is lowered to 26 feet. Find the number of units of work done in the operation, the weight of a gallon of water being 10 lbs. (S. and A. Exam. 1887.) For an answer to the first part of this question refer to the previous part of this lecture. Diagrams of Work.-(1) Against a Uniform Resistance.-If the resistance overcome is uniform, then the work done may be graphically represented by the area of a rectangle. To find the work done in overcoming a uniform resistance of 5 lbs. through a distance of 10 ft.: Plot down a vertical line to any convenient scale to represent P (or 5 lbs.) and a horizontal line to the same scale to represent L (or 10 ft.) The area P × L or 5 × 10= Plbs. --10ft. DIAGRAM OF UNIFORM WORK. Then complete the rectangle. 50 ft.-lbs. of work. In the accompanying figure a scale of inch has been used to represent both 1 lb. and I ft., consequently each of the small squares represents to scale one foot-pound of work. I (2) With a Uniformly Increasing Resistance. If the resistance uniformly increases - for ex ample, in the raising of a length of rope or chain vertically by one end from the ground, then the work Plbs. done may be graphically represented by the area of a right-angled triangle, -.-Lft. where P represents the total weight DIAGRAM OF Work for an of chain in lbs., and L its total length in feet. INCREASING RESISTANCE. Here the work done per foot of length of chain lifted, uniformaly increases from a minimum to a maximum, until the whole rope or chain is off the ground. When any known length, 7, has been lifted, then the area enclosed by the triangle whose horizontal side is l, and vertical side p represents the work done. (3) With a Uniformly Decreasing Resistance.-If the resistance Plbs. -Lft. DIAGRAM OF WORK FOR A uniformly decreases, as in the case of winding a rope or chain upon the barrel of a winch or crane, then the work done will also be represented graphically by the area of a rightangled triangle, where P represents the total weight of rope or chain in pounds being lifted at the start, and L its length in feet. fere the work done per foot of length of chain lifted, gradually diminishes from a maximum at the start to a minimum, when the last foot is being lifted. As in Case (2), you can at any time know the work done or still to be done from the scale diagram, if you know the length of chain lifted or to be lifted. For example, if 7 feet have still to come on to the barrel, then the vertical ordinate p on the scale diagram will represent the pull being exerted at the time, and consequently the work still to be done. Or, generally, with any gradually increasing or decreasing resistance the work done is equal to the mean of the average resistance in lbs. x the distance through which it acts in feet. (4) With a Combination of Uniform and Variable Loads.-When one part of a load is uniform and another part variable, as in the case of lifting a weight with a chain, by winding the chain on the barrel of a winch or crane, the diagram of work for the uniform load is naturally a rectangle, and for the chain a triangle if the chain is completely wound on to the barrel, or a trapezoid if there is still some portion of it to be lifted.* 1 1 P DIAGRAM OF WORK FOR A *See p. 5 of the Author's "Elementary Manual on Steam and the Steam Engine" for how to find the Area of a Trapezoid. Let P L = the uniform pull required to lift the load or overcome the uniform resistance. the distance the weight is lifted. the weights of chain hanging at the commencement and at the finish of the lift. Work done in lifting the uniform load = P x L Work done in lifting the variable load = P1 +P2 × L ...Whole work=PxL+P1 + 12 × L= (P + 2 The diagram DEFC represents the work done and also the variation of the resistance during the lift. The rectangle ABCD represents the work done in overcoming the uniform load, and the trapezoid ABFE the work done in overcoming the variable load. The resistance at any instant of the lift will be represented by the vertical line drawn from the horizontal base DC to the inclined line EF through the point on DC or AB which represents the position of the load at that instant. The part of this vertical line intercepted between AB and EF will represent the resistance offered by the variable part of the load at the instant considered. Thus, at the commencement of the lift the total resistance is P+p, and represented by DE, at the end of the lift the total resistance is P+p, and represented by CF. At 1,, and of the lift the total resistances are represented by the vertical lines GH, KL, and MN respectively, while the resistances due to the variable part of the load at these points are represented by the lengths gH, kL, and mN respectively If the final resistance due to the variable part of the load was zero (as would be the case if the whole of the chain were wound on to the barrel) then the diagram of work for this part of the load would be the triangle AEB. EXAMPLE VI.-Explain fully the mode of measuring the work done by a force. What unit is adopted? A weight of 2 cwts. is drawn from a mine, 30 fathoms deep, by a chain weighing 1 lb. per linear foot; find the number of units of work done. (S. and A. Exam. 1893.) Also find the resistance offered when the weight has been raised through 1, 1, and 2 of the whole depth respectively. ANSWER. (1) The work done by a force is measured by the product of the force into the distance through which that force moves in its own direction. If P be the force in pounds, and L the distance in feet through which it moves in its own direction, then, Work done = P x L ft.-lbs. (2) The unit of work adopted in this country is the work done when a force of one pound is moved through a distance of 1 foot, and is called the foot-pound (ft.-lb.). = (3) Referring to the previous figure, make AB to represent 30 x 6 = 180 ft., the depth of the mine, AD to represent 2 × 112 224 lbs. the weight of material raised, AE to represent I X 180 = 180 lbs. the weight of chain at beginning of lift. Then assuming the whole of the chain to be wound up, complete the rectangle ABCD, and join E and B. The area of the figure DEBC then represents the work done. .•. Work done = area DEBC = { (DE + CB) × AB. But DE= DA+AE=224 + 180=404 lbs. CB=DA=224 lbs., AB= 180 ft. Work done = (404 + 224) × 180 ft.-lbs. = 56,520 ft.-lbs. (4) The resistance at lift, or when the weight has been raised 45 ft., is Gh=Gg+gh=224 + 2 x 180 = 359 lbs. At lift the resistance is Kl = Kk + kl = 224 + × 180 = 314 lbs. At lift the resistance is Mn = Mm+mn= 269 lbs. 224+1 × 180= Power or Activity is the rate of doing work.*-In estimating or testing the power of any agent the time in which the work is done must be noted and taken into account. Consequently, we speak of the activity or power of a man, of a horse, or of an engine, as capable of doing so many foot-pounds of work per minute. Units of Power.†-The unit of power adopted in this country is called the horse-power. It is the rate of doing work at 33,000 ft.-lbs. per min., or 550 ft.-lbs. per sec., or 1,980,000 ft.-lbs per hour. The Horse-power Unit was introduced by James Watt, the great improver of the steam engine, for the purpose of reckoning the power developed by his engines. He had ascertained by experiment that an average cart-horse could develop 22,000 footpounds of work per minute, and being anxious to give good value to the purchasers of his engines, he added 50 per cent. to this amount, thus obtaining (22,000 + 11,000) the 33,000 foot-pounds per minute unit, by which the power of steam and other engines has ever since been estimated. *The word power is very frequently misapplied by writers and students, for they often call the mere pull, pressure, or force exercised on or by an agent the power. Students should strenuously avoid this misuse of the word power, and never employ it in any other sense than as expressing a rate of doing work, or activity. + In Electrical Engineering the Unit of Power is called the Watt, and it equals 107 ergs per second, or 746 Watts = I horse-power. |