parallel, and in the same plane, and the arm A C should be equal to the arm A B. Now if A, B, and C are all in the same straight line, and if the weight P is greater than the weight Q, the balance will tip entirely over unless some counteracting force is available. This is found in the weight of the balance beam itself which is so adjusted that its center of gravity does not lie exactly on the edge A, but slightly below it, say a distance x. Then when P is greater than Q, if I is the length of the balance arm and w the weight of the beam which has W FIG. 32. turned through a small angle a, the moment of the force P will be Pl cos a counter-clockwise, while the moments of Q and of w will be Ql cos a and wx sin a clockwise, and we have therefore in equilibrium or Pl cos a=Ql cos a+wx sin a wx P-Q= tang a. which result shows that the deflection of the balance is proportional to the difference of the two weights, and that to make the deflection large when P-Q is small, the weight of the balance beam w must be small, also x must be small in comparison with l. 70. Double Weighing. If the arms of a balance are not of equal length, the true weight of a body may be obtained by the method of substitution, in which the body is placed in one pan of the balance and some convenient counterpoise, such as sand or shot, which will exactly balance it, placed in the other pan. The body is then removed and weights substituted for it until equilibrium is again reached, the substituted weights are then equal to the weight of the body. Or the method of double weighing may be employed. Let r be the length of the right arm of the balance and I the length of the left arm, then if a body whose real weight is P is placed in the right pan and balanced by weights W in the left pan, we have by the equality of moments Pr=Wl. Then interchanging body and weights, it is found that a weight W' is required to balance it when it is in the left pan, which gives Pl=W' r. Multiplying the two equations together we have Plr WW'lr_or P= Problems. 1. At what point on a pole must a weight of 50 lbs. be hung so that a boy at one end may carry as much as the man at the other end? And how much does each carry? 2. If the pole itself is uniform and weighs 10 lbs., where must the 50-lb. weight be hung so that the man may carry twice as much weight as the boy? 3. A beam 20 ft. long is carried by three men, one at one end and the other two supporting it between them on a cross-bar at such a point that each man carries an equal weight. Find where the cross-bar must be placed. 4. Find the center of gravity of a uniform bar 3 ft. long weighing 6 lbs. and having a 2 lb. weight on one end and a 5 lb. weight on the other. 5. Forces 2 and 4 acting upward are applied to a horizontal bar at 2 ft. and 3 ft. from the left-hand end, respectively, also forces 3 and 1 acting downward are applied at 1 ft. and 5 ft. from the same end. Find amount and point of application of a single force producing equilibrium. 6. A board 2 ft. square is acted on by five forces applied at the same points as shown in figure 22, but the forces instead of being 2, 1, 2, 3, 2, beginning at the top, are 3, 3, 1, 2, 4, respectively. Find the direction and amount of the force needed to produce equilibrium and how far from the center of the board its line of action must lie. 7. A ladder standing 6 ft. from a smooth vertical wall rests against it at a point 30 ft. from the ground. If the ladder weighs 60 lbs. and its center of gravity is of its length from the bottom, find the force with which it presses against the wall, also the amount and direction of its pressure against the ground; that is, find the vertical and horizontal components of the pressure. Note. The force between ladder and wall must be perpendicular to the latter if there is no friction between them. 8. When is the ladder in problem 7 more liable to slip, when a man is near the top or bottom, and why? WORK AND ENERGY. 71. Work. When a body acted upon by a force moves in the direction in which the force is acting, work is said to be done. When the body yields to the force, work is said to be done by the force or upon the body; but when the body moves against the force, work is done by the body or against the force. The amount of the work done is measured by the product of the force by the distance which the body moves along the line of action of the force. Thus when a 2-pound weight is raised 3 feet, it moves & distance of 3 feet against a force of 2 pounds, and therefore 6 foot-pounds of work is done against the force of attraction of the earth. the If the motion of the body is not in line with the resultant force, then in estimating work only that component of the motion which is in the direction of the force is to be taken into account. For instance, in raising a barrel into a wagon the work done is the same whether the barrel is lifted directly from the ground or rolled up an inclined plane. For the weight of a body is a force that acts vertically downward, consequently in estimating work done against weight, only the vertical distance through which the body is moved is to be considered. The work done in raising a weight or compressing a spring is the same whether done in a second or in an hour. The time required to do the work determines the rate of working, but has nothing to do with the amount of work. It is remarkable that although force and distance are both vector quantities, work, which is their product, is not a vector quantity. It has nothing to do with direction, and consequently to get the total work done upon a body by several different forces, the work of each may be reckoned separately and then the sum taken. Motion is essential to work. A great weight may rest on a support, but no work is done in supporting the weight though a great force is exerted. 72. Rate of Working. Horse-power.-A given amount of work may be done either in a short time or a long time, and in commercial operations the rate of working, or the work done per second or per hour, is an important consideration. Thus in case of an engine we wish to know how much work it can do in a given time, and its rate of working is known as its power. Power may be measured by the number of grams weight that can be raised one centimeter in one second, or by the number of pounds that can be raised one foot per second; but the unit of power introduced by James Watt and commonly used in engineering practice is the horse-power (written H. P.). One horse-power-550 foot-pounds per second, or 33000 foot-pounds per minute. That is, a 10 H. P. engine can raise 330 lbs. through a height of 100 ft. in one-tenth of a minute, or 3300 lbs. through a height of 10 feet in the same time. Power is also measured in ergs per second, and watts. these units will be discussed later (§§ 104, 654). 73. Energy. The importance of the idea of work lies in the fact that a body upon which work is done acquires thereby capacity to do an equal amount of work in returning to its original state. The capacity to do work is called energy. Thus work is done when a spring is bent, and the spring acquires energy which is measured by the work that it can do as it unbends. Also 10 lbs. weight raised 100 ft. above the earth has had 1000 ft. lbs. of work expended in raising it, and it has gained the power to do that same amount of work in returning to its original position. The energy of a bent spring resides in the spring itself in virtue of its internal stresses; but in case of the raised weight the energy belongs not to the weight alone, but to the system of two bodies, the earth and the weight, which are separated in opposition to the stress or attraction between them. 74. Kinds of Energy. In both illustrations given above the energy depends on the relative positions of bodies or parts of bodies between which there exist stresses. There is another form of energy which depends not upon stress, but upon the motion of matter. Suppose the raised weight is set free and allowed to fall with nothing to resist it, the force of the earth's attraction is exerted upon the mass as it falls and consequently work is done and energy expended, but in this case the work is all spent in giving velocity to the falling mass. When the weight reaches the bottom it has lost all its advantage of position, but it still has power to do work in virtue of its motion, and the work it can do before coming to rest is exactly equal to the work that was done upon it in giving it motion. The mass therefore still retains the energy that it had in the raised position, but it is now energy of motion. The energy which a body or system of bodies has in virtue of stresses is called potential energy. The energy which a body has in consequence of the velocity of its mass is called its kinetic energy. 75. Illustration. If a mass is hung so that it can freely swing as a pendulum, when it has been raised to the position A (Fig. 33) it has been raised through the vertical distance h from B to D, and therefore has at A an amount of potential energy more than at B by the work done in raising it from B to A. If allowed to fall freely it will reach the bottom, moving with sufficient velocity to carry it up to C on the same level as A. At the bottom the mass has energy of motion or kinetic energy. It has entirely lost the advantage of position which it had at A, the work done in D h B raising it to A being now wholly transformed into energy of motion. But as the mass rises from B toward C it loses velocity for it is doing work and using up the store of kinetic energy that it received in falling, changing it again to potential energy. The pendulum has thus a constant store of energy which changes back and forth from one form to the other, the sum of the two always being constant, except as energy is gradually lost through friction and air resistance. 76. Work against Friction.-There is one case, however, in which the work done upon a body does not seem to increase its energy or power to do work. When a weight is pushed from one point to another on a level table force has to be exerted to overcome friction. The weight, however, remains at the same level above the earth and has no more power to do work in the new position than before it was moved. The work expended seems to be quite lost. But investigation has shown ($409 et seq.) that whenever work is done against friction heat is developed in amount exactly proportional to the work done; and also that when work is obtained from a heat engine a precisely corresponding amount of heat disappears. It is therefore concluded that the work |