n 



APPLIED MECHANICa 



[MENSURATION or BOM us. 



luue U a recUngle 4 feat long by 3 feet wide, and whoso 

 height U 2 feet inches. 



Area of bate A X > X H height - 30 cub. foot, or 

 30 x 6-2321 - 187 gala, nearly. 



impU 2. Required tlio so'id contents of a roller 

 2 in*, diameter and 42 ins. ! 



I'-' x "' X 7S"4 X 42 169G5-6 cubic inches, 

 i the priim or the cylinder U hollo*. 



>. Subtract the area of the internal part of the 

 bae from that of the whole base, and multiply by the 

 height. 



impte 3. Required the weight of a cast-iron hollow 

 cylinder 6 feet high and 3 ) inches diameter internally, 

 ;ig metal 1 inch thick (reckoning 3 '8 cubic inches of 

 cast-iron equivalent to lib.) The external diameter is 

 the internal dimmer increased by twice the thick- 

 nan of metal ; it U, therefore, 32 inches, and its area 

 would be 32 X 32 X 7854. The area of the internal 

 would be 30 X 30 X 7854 ; and, subtracting, we have 

 (: x 32 30 X 30) X 7854 = 97 '430 sq. inches for 

 the area of the section of metal. The solid contents are 



5846-16 

 97 436 X 60 ins. = 5846 16 cub. ins., and jj^j =1538 '46 



Ibs., or 13 cwt. 2 qrs. 2Gi Ibs. 



When the diameter of* a hollow cylinder is large in 

 proportion to the thickness of its material, as in the case 

 of a boiler, which may be several feet in diameter, while 



the thickness of its metal is less than half an inch, we 

 may multiply tho area of its surface by the thickness of 

 material to ascertain the quantity of material In this 

 calculation tho following points inuit be observed. 



B lilors are generally made of wrought-iron plate ; and 

 it happens that a square foot of iron plate, Jth of an 

 inch thick, weighs 6 Ibs. In order to ascertain the 

 weight in pounds of a wrought-iron boiler Jth of an inch 

 thick, we should then have to multiply its surface by 5 ; 

 and were the thickness gths or J th, we should multiply 

 by 2 X 5 or 10 ; for thickness f ths, multiply by 3 X 5 

 or 15, and so on. But iu constructing boilers, there are 

 many plates used, which overlap each other at their 

 points ; and at the comers there are angle irons, and 

 numerous rivets are used to make the joints tight. 

 Making allowance for all these additions to the weight, 

 we may estimate the weight per superficial foot at about 

 C Ibs. for every Jth of an inch in thickness, or 3 Ibs. for 

 every ^th of an inch. Again, as T-'T U nearly jUth, we 

 may roughly approximate the weight in cwts. oy the 

 following rule : 



Rale. Multiply the surface (in square feet) by the 

 thickness (in ths of an inch), and divide by 20 for the 

 weight (iu cwts.) 



Example 4. Required the weight of a Cornish boiler, 

 external diameter 4 feet, diameter of flue-tube 2 feet, 

 length 12 feet, thickness J and fa of an inch (or 3$ eighths 

 of an inch). 



External circumference . . 4 X 3f 

 Circumference of flue . . . 2x3} 



Both . . 



Area of end of boiler 

 Area of flue . . . 



nearly length inrface of curing and floe. 



6 X 3f = 18-9 X 12 = 226-8 



Difference 



4 x 4 x 7854 

 2 X 2 x 7854 



^^^ rarftct of cndi. 



12 x 7854 = 9-4 X 2 (ends) = 18-8 



And 



243-6 X 

 20 



43 cwt. nearly, or 2 tons 3 cwt. 



245-0 total surface. 



The enbic contents of a sphere are Jrds of that of a 

 cylinder of the same diameter and altitude. But 'the 

 altitude being equal to the diameter, and jjrds of 7854 

 being -5236, the contents of the sphere may be found by 

 tho following method : 



Ride. Multiply the cube of the diameter by -5236. 



E.mmple 5. Required the number of cubic inches in 

 a ball 12 inches diameter. 



12 X 12 x 12 X '5236 = 904 7808 cubic inches. 



When the sphere is hollow, 



Rule. Subtract the cube of the internal diameter 

 from the cube of the external diameter, and multiply 

 by -6236. 



Example 6 Required the weight of a hollow ball of 

 iron, 30 inches diameter externally, metal 1 J inch thick, 

 reckoning 38 cubic inches to the pound. 



External diameter 30 X 30 X 30 = 27000 cubed 

 Internal diameter 27 X 27 X 27 - 19683 



Difference ..... 7317 



And 



7317 X "5236 

 38 



= 1008 Ibs. or 9 cwt. 



The cubic contents of a cone are Jrd of that of a cylin- 



>l.-r of equal base and altitude. Hence, to find the con- 

 tent* of a cone : 



-Multiply the square of the base by the height 

 and by -201 8. 



EfmpleT. Required the solid contents of a cone, 

 having a base 18 inches diameter, and a height of 22 



I:. 



18 X 18 X 22 X -2C18 - 1866-11 cubic inches. 

 The solids with which the practical mechanic has to 

 deal are chiefly those called KoluU of revolution, or such 

 M may be conceived to be produced by the revolution of 

 Bailie figure round an axis. Any work that is formed in 

 a turning. lathe U a solid of revolution. An imaginary 



line extending between the centres of the lathe is the 

 axis ; and if the solid were cut across by a plane passing 

 through this line, the section or surface exposed on the 

 removal of the half cut off, is symmetrical on each side 

 of the axis or centre line. One-half of this symmetrical 

 figure, or tho part lying on either side of the axis, is 

 called the generating figure. Thus, a cylinder is gene- 

 rated by the revolution of an oblong or rectangle round 

 one side. A cone is generated by the revolution of a 

 right-angled triangle round its perpendicular. A sphere 

 is generated by the revolution of a semicircle round its 

 diameter. A spindle, by the revolution of a segment of 

 a circle, or other curve, round its chord. In the same 

 way, other mathematical figures, such as the ellipse, or 

 the parabola, are capable of producing solids of revolu- 

 tion by causing them to revolve round their axis. When 

 the ellipse is supposed to revolve round its shorter axU, 

 the solid produced is called an oblate or shortened 

 spheroid ; when it revolves round its longer axis, the 

 solid is an oblong spheroid. The parabola, by revolu- 

 tion, produces the paraboloid. 



There are singular relations among some of these 

 solids of revolution, one of which it is useful to remem- 

 ber. If A B E D (Fig. 271) represent the section of 

 a cylinder, -whoso base has a diameter AB double its 

 height, or axis, C F A F B the section of a cone on tho 

 same base, and of the same height with the cylinder 

 A G F H B the section of half a sphere, and A K F L B 

 the section of a paraboloid ; then 



The iolid content* of the cone are Jrd of that of the cylinder. 

 ,, ,, hemisphere are Jrdf ,, ,, 



paraboloid are i 



As the paralx>loid comes midway between the cone and 

 the hemisphere in form of section, so its capacity i 

 midway between them in amount. In i 's of 



mensuration of solids of revolution, when tho form of 

 lection is something between tli.it of a cone and of a 



