1. Find the multiplier for changing revolutions per minute into radians per second. 2. How soon after VIII. are the hour and minute hands directly opposite to each other? 3. Express in degrees, grades, and radians the angle made by the hands of a watch at 3.30 o'clock. 4. The minute and second hands of a watch point in the same direction at XII. When do they next point in the same direction? 5. At what time after n o'clock is the minute hand first ten minutes before the hour hand? What is the greatest value of n to allow this to happen within the hour? 6. Two clocks are together at XII.; when the first comes to I. it has lost a second; and when the second comes to I. it has gained a second. How far are they apart in 12 hours? 7. Two clocks are correct at mid-day; when the first clock indicates VI. in the afternoon, the second wants a minute to VI.; and when the second indicates midnight, the true time is 2 minutes past XII. What does the first clock indi cate at midnight? They meet for the Where will they 8. Two men walk opposite ways round a circular course. first time at the north point, the sixth time at the east point. meet for the sixteenth time-and what are their relative speeds? 9. A clock loses at the rate of 8.5" per hour when the fire is alight, and gains at the rate of 5'1" per hour when the fire is not burning; but on the whole it neither loses nor gains. How long in the 24 hours is the fire burning? 10. In going 120 yards the forewheel of a carriage makes six revolutions more than the hindwheel. If each circumference were a yard longer, it would make only four revolutions more. Find the circumference of each wheel. 11. Two men start together to walk round a circular course, one taking 75 minutes to the round, the other 90. When will they be together again at the starting point? 12. The hour hand of a watch is 3/7 of an inch long, the minute hand 4/5 of an inch, and the second hand 1/3 of an inch. Compare the linear speeds of their points. 13. Deduce the equivalent of longitude for one minute of time, and for one second of time. 14. What is the circumferential speed of a wheel 28 feet in diameter when making five revolutions per minute? 15. The diameter of the earth is nearly 8,000 miles; required, the circumference of the earth at the equator, and the number of miles per hour which the inhabitants of latitude 60° are carried by the earth's diurnal rotation. 16. What is the velocity due to the earth's rotation of a person dwelling on the 45th degree of latitude? 17. The front wheel of a bicycle is 52 inches in diameter, and performs 5,040 revolutions in a journey of 65 minutes. Find the speed in miles per hour at which it has travelled, assuming the ratio of the circumference to the diameter of a circle to be as 22 to 7. 18. When a steamer sails due west, at the latitude of 45°, at the rate of 14 knots per hour, what is the rate at which the clock gains time? 19. A reaping machine works round a rectangular field of grain 357 yards by 216 yards at the average speed of 3 miles an hour, the breadth cut by the reaper being 5 feet. How long will it take to cut down the field? 20. Find the distance traversed in ploughing 12 acres of land when the furrow is cut 11 inches broad; also the time required when the horses move at the average speed of 2 miles an hour. SECTION XX.--RATE OF CHANGE OF SPEED. ART. 115.-General Unit. By acceleration is meant rate of change of velocity with respect to time. To each of the varieties of velocity discussed there is a corresponding variety of acceleration. Rate of change of speed and rate of change of velocity are both expressed in terms of (L per T) added per T, the only difference being that in the former case we have not, and in the latter we have to consider the direction of L. A peculiarity of this rate is that time enters twice as the independent quantity. In order to be independent of one another, the latter interval must elapse before the former. The unit of the British absolute system is the foot per second per second; and that of the C.G.S. system is the centimetre per second per second. ART. 116.-Rate of Change of Speed. Here our attention is entirely restricted to one path. If the increment to the speed has the same sense as the existing speed, it is said to be an acceleration proper; if it has the opposite sense, it is said to be a retardation. Suppose that the point is at one instant moving with a speed of v L per T, and that after an interval of n T it is moving with a speed of v L per T; then the change of speed in the course of the interval is v2 – v1 L per T. If this change has been made uniformly, then the rate of change during the interval is ART. 117.-Application of Rate to find Space passed over. Given that a point is subject to a constant rate of change of speed, a L per T per T, if the interval during which it goes on is t T, then the change of speed is ta L per T If the subsequent interval is t' T, then t'ta L is the distance gone over due to the speed which was imparted before the interval began. Hence the rate may be viewed in the form of an equivalence Fig.14. (Art. 81), we shall get the correct result by supposing the average speed to have existed throughout the interval (Fig. 14). The Hence, instead of the previous equivalence, we have Ja L = T We have T2 because the two intervals are identical. The comes in for the same reason that it appears in the equivalence for the area of a triangle. ART. 118.-Derived Rate. Given that a point is subject to a constant rate of change of speed イ T, (1) then, as we have shown, the ratio of space described due to that acceleration is 1a L = T2 By squaring the first equivalence we obtain a2(L per T)2 T2 ; = and by eliminating T2 between (3) and (2), we deduce (2) (3) (4) We shall afterwards find (Art. 159) that 2(L per T)2 is the unit for expressing the kinetic energy of unit mass. ART. 119.-Composition of Effects. Let the initial distance be s L, the initial speed v L per T, and the constant rate of change of speed a L per T per T. As all the quantities are supposed to be along the same line, the composition is effected by simple addition; but the quantities may differ in sign. Hence, after t T, the velocity will be v + at L per T ; and the distance will be s+vt + 1ať2 L. EXAMPLES. Ex. 1. Assuming 32.2 as the foot-second measure of the acceleration produced by gravity, express the same quantity numerically in terms of the mile-hour unit. |