3. On attaching a spring scale to a block of wood and pulling it over a floor, the reading of the scale was found to be 16 kg. How much work was done in drawing the block over a distance of 22 m.? 4. A mass of 8 kg. is raised 3 m. vertically at New York. How many joules of work are done? 5. How much work is done in lifting a barrel of flour weighing 196 lb. from the ground into a wagon 3 ft. high? 6. A man weighing 186 lb. carries a package weighing 32 lb. upstairs from the first to the third floor of a building. How much work does he do if the vertical distance is 19 ft.? How much does he do upon himself? How much in carrying the package? 7. A steam engine is used in hoisting coal from a mine 147 ft. deep. What must be the horse power of the engine to raise 380 long tons in 8 hr., if friction increases the load 12 %? What is the power of the engine in watts? 8. What is the potential energy of the head of a pile driver weighing 500 lb. when it is 28 ft. above the end of the pile? When it is 16 ft. above the end of the pile? If it is let fall from the height of 28 ft., it will strike the pile with a velocity of 42.438 ft. per second. What will be its kinetic energy on striking? 9. A quarry crane hoists a block of marble 6 ft. long, 4 ft. wide, and 2.5 ft. thick. How much work is done in raising the block 12 ft. if the marble weighs 160 lb. per cubic foot? Neglecting friction, what must be the horse power of the motor to do this in 3 min.? What is the potential energy of the block at that height? 10. What is the kinetic energy of a 250-lb. cannon ball that has a velocity of 1260 ft. per second? 11. What is the kinetic energy of a freight car and load, the whole weighing 40,000 lb., when moving at the rate of 20 mi. per hour? 12. An elevator weighing 1800 lb. more than its counterweight carries a load of 6 people of an average weight of 140 lb. each. Neglecting friction, what must be the horse power of a motor that will lift it 72 ft. in 30 sec.? 13. An ordinary brick weighs 5 lb. How long would it take to fall to the ground from the top of the Kodak Company's chimney in Rochester, which is 366 ft. high? With what velocity would it strike? What would be its kinetic energy on striking? III. GRAVITATION AND GRAVITY 79. Law of Universal Gravitation. Gravitation is the name given to the mutual attraction between different bodies of matter. The matter considered may be two books lying on a table, or two stars separated by millions of miles. The attraction is universal, and the Law of Universal Gravitation may be stated as follows: Every particle of matter in the universe attracts every other particle with a force that varies directly as the product of the masses of the particles and inversely as the square of the distance between them. This leads to the formula which is applicable to all mutual attractions, namely, F. = Mm (24) in which a is the unit of attraction; i.e., the attraction between two units of mass at a unit's distance. For comparing two attractions of the same kind we may write the proportion The momenta given by mutual attraction to the two bodies between which the attraction acts, are equal. A man standing in a rowboat and pulling on a rope that is fast to a sloop moves the boat faster than the sloop, but only because its mass is much less. The momentum imparted to the sloop is equal to that given to the rowboat. 80. Gravity. While the term gravitation is applied to the universal attraction existing between particles of matter, the more restricted term gravity is applied to the attraction that exists between the earth and bodies upon or near its surface. The law given above applies to gravity, provided that d is measured in a straight line from the center of the earth to the center of mass of the body. This line is called a vertical line, or sometimes a plumb line (from the Latin word plumbum, which means "lead"), as vertical lines are frequently determined by suspending a mass of lead, the plumb bob, at the end of a cord (Fig. 50). 81. Weight. The weight of a body is the measure of the mutual attraction that exists between the earth and that body. This force is the resultant of the attractions between the earth and all the particles of the body. The weights of any two bodies at the same place are proportional to their respective masses. FIG. 50 Since the polar diameter of the earth is 26 miles less than the equatorial, it is evident that the weight of a body will vary with the latitude as well as with the elevation above the sea level. The weight of a body carried from either pole toward the equator is decreased by the increase in its distance from the center. There is also an apparent decrease, owing to the increase in the centrifugal force of the earth's rotation. Bodies on the equator move with a velocity of more than a thousand miles per hour, and the centrifugal force there is of the force of gravity, while at the poles it is zero. Should the earth rotate 17 times as fast as it now does, the centrifugal force would equal the force of gravity, since centrifugal force varies as the square of the velocity (Formula 17). 82. Weight above the Surface. of a body is at the surface of the earth. 2 The maximum weight If a body is removed above the sea level, as on the top of a mountain, or in a balloon, the distance d between it and the center of the earth is increased, and its weight is diminished. The relation between weight at the surface and weight above the surface may be expressed by the proportion in which W is the weight at the surface; w, the weight above the surface; D, the distance from the center to the surface of the earth; and d, the distance of the body from the earth's center. 83. Center of Gravity. The attraction of gravity on any body tends to draw its particles toward one point, and hence, strictly speaking, the directions of these forces are not parallel. As the radius of the earth is very large, however, compared with the size of any object which is weighed, their divergence from parallel lines is, practically, not measurable. The point of application of the resultant of all the parallel forces (§ 62) that make up the weight of a body is. its center of gravity, center of mass, or center of inertia. Demonstration. Fit in a small wooden handle (or in a fixed support), two wires (Fig. 51): one, A, straight and the other, B, bent twice a right angles. In a piece of thin board C of any shape bore holes D and E in two corners. Suspend the board by one of these holes D from the wire B, and from A suspend a plumb line. See that D is exactly halved by the plumb line when at rest, and mark a point F opposite the line. Suspend the board from the hole E, and mark the point G. Draw lines DF and EG, and their intersection O will determine the center of gravity. Test the accuracy of the work, by making a hole at O and rotating on the end of A. Find, in the same way, the centers of gravity of a triangle, a square, a rectangle, and a circle. In the above cases the center of gravity is midway between the two surfaces at the point 0. It would still be at O, if the thickness of the board were infinitely reduced; hence we may speak of the center of gravity of a surface. The center of gravity of any body may be found by suspending it successively from two points on the body and finding the intersection of the lines of direction from those points of support to the center of the earth. This is because a body suspended from any point will hang with its center of gravity vertically below the point of suspension. The center of gravity is frequently outside the substance of the body, as in the case of a ring. A 84. The Center of Gravity of a Number of Bodies rigidly connected may be determined by considering the weight of each body as a parallel force applied at its center of gravity, and then finding the point of application of the resultant of these forces (§ 62). Suppose three parallel forces, P, Q, and S, to be applied at three points, A, B, and C, rigidly connected, as in Fig. 52. C E D B R' R FIG. 52 The resultant R' of P and Q will equal P+Q, and its point of application will be at a point D, determined by the proportion (§ 62) P: R' DB: AB, = |