8. The drum (on which the string winds) of a windlass has a radius of 3 inches, and is turned by spokes 2 feet long. If we wish to raise a weight of 100 pounds, what force must we exert at the end of a single spoke? 9. Figure 52 illustrates the ordinary form of safety valve on a steam engine boiler, consisting of a bar of iron hinged at one end F and weighted at the other end, and resting at some point between on Fig. 51. a piston which sits upon the steam. If the area of the end of the piston (where it rests on the steam) is 16 square inches, what weight W must be hung on the end 100 Ibs. so that when the steam pressure p has risen to 100 pounds per square inch the piston will rise so as to let some of the steam escape? 10. In any of the levers illustrated in Ex. 5, the distance D from the weight to the fulcrum is called the weight arm, while the distance d from the power to the fulcrum is called the power arm. Show that the general law of the lever (see Ex. 5) may be stated in the following form, which is the way it is usually remembered by engineers: The weight times the weight arm equals the power times the power arm. 113. Gear Wheels. In the figure one gear wheel is turning another in the usual way. If T is the number of teeth on the larger wheel and N is the number of turns this wheel is making per minute, while t is the number of teeth on the smaller wheel and n is the number of turns this wheel is making per minute, then we have the following law of gear wheels: EXERCISES – GEAR WHEELS 1. If the large wheel in Fig. 53 has 60 teeth and is making 100 revolutions per minute, how many revolutions per minute is the smaller wheel making if it has 20 teeth? [Hint. By the law stated above, we have 60/20 = n/100.) 2. A wheel of 90 teeth that is making one revolution per second, fits into a smaller wheel which is revolving twice as fast. How many teeth are there in the smaller wheel? 3. In order that the small wheel may always revolve just twice as fast as the large one, how should the wheels be made? Why? [Hint. The example supposes simply that n=2 N.] 4. A certain gear wheel of T teeth is making N revolutions per second. It fits into another similar wheel having t teeth. Show that if the latter wheel be replaced by one having r more teeth, the new one will revolve slower than the old one by the amount TNr rev. per sec. t(t+r) 114. Belts. In Fig. 54, one wheel is turning another by means of a belt in the usual manner. If D is the diameter of the larger wheel and N is the number of turns this wheel is making per minute, while d is the diameter of the smaller wheel and n is the number of turns this wheel is making per minute, then we have the following law of belts: = , or DN=dn. Fig. 54. EXERCISES — BELTS 1. Show that if the two wheels are of the same size, each makes the same number of turns per minute. [Hint. This supposes D=d.] , 2. Show that if the diameter of the small wheel is one third that of the large one, it will make three times as many turns per minute. 3. A wheel whose diameter is 10 inches is revolving at the rate of 1 turn per second and is belted to a smaller wheel whose diameter is 3 inches. By how many turns per minute will the smaller wheel be slowed down if its diameter be increased by 2 inches? 4. A wheel of diameter D is making N revolutions per minute, and is belted to another wheel whose diameter is d. Show that in order to increase the speed of the latter wheel by q revolutions per minute, it would be necessary to diminish its diameter by DNg 115. Similar Figures. When two geometric figures have exactly the same shape (though not necessarily the same size) they are called similar figures. Thus, any two circles are similar figures, as likewise any two squares, or spheres, or cubes. C Two triangles may be similar, as illus trated in the figure. Note that, though not of the same size, these triangles have exactly BÁ B the same shape. Fig. 55. The following facts are shown in Geometry to be true of any two similar figures : (a) Corresponding lines are proportional. Thus, in the two similar triangles above, if the side A B of the one triangle is twice as long as the corresponding side A'B' (read A prime, B prime) of the other triangle, then BC is twice as long as B'C'; that is, AB - BC. A'B' B'C In the same way, we have also to (b) Areas are proportional to the squares of corresponding lines. Thus, if one circle has a radius of length R and another circle has a radius of length r, the area A of the first circle is to the area a of the second as R2 is to ra; that is, we have the proportion A – R2. Compare Ex. 15, p. 182. (c) Volumes are proportional to the cubes of corresponding lines. Thus, if one sphere has a radius of length R and another sphere has a radius of length r, the volumes V and v of the spheres are such that we have the proportion V/v = R3/r3. EXERCISES —SIMILAR FIGURES 1. In the two similar triangles shown in § 115 suppose AB=1 foot, A'B'=8 inches, and BC =17 feet. How long must B'C' be? (Hint. Measure all lengths in inches and let x be the length of B'C'. Then by (a), § 115, we have 12/8 = 18/x.] 2. If a tree casts a shadow 50 feet long when a post 4 feet high casts a shadow 5 feet long, how high must the tree be? 3. A triangle has its sides 3 inches, 4 inches, and 5 inches long. Another triangle of the same shape has its shortest side 2 inches long. What are the lengths of the other sides of this triangle? 4. Of the two triangles in Ex. 3, the first has an area of 6 square inches. What is the area of the second ? [HiNt. Let x be the area of the second. Then, by (b) of $ 115, we have 6 32 22' that is, 5. Compare the areas of two city lots of the same shape if a side of the one is twice as long as the corresponding side of the other. Does your answer apply no matter what the shape is so long as it is the same for each lot? |