Machinery''''s Handbook 27th Episode 1 Part 4 pps

RADIUS OF GYRATION 235 Sphere: Hollow Sphere and Thin Spherical Shell: Ellipsoid and Paraboloid: Center and Radius of Oscillation.—If a body oscillates about a horizontal axis which does not pass through its center of gravity, there will be a point on the line drawn from the center of gravity, perpendicular to the axis, the motion of which will be the same as if the whole mass were concentrated at that point. This point is called the center of oscillation. The radius of oscillation is the distance between the center of oscillation and the point of suspension. In a straight line, or in a bar of small diameter, suspended at one end and oscil- lating about it, the center of oscillation is at two-thirds the length of the rod from the end by which it is suspended. When the vibrations are perpendicular to the plane of the figure, and the figure is sus- pended by the vertex of an angle or its uppermost point, the radius of oscillation of an isos- celes triangle is equal to 3 ⁄ 4 of the height of the triangle; of a circle, 5 ⁄ 8 of the diameter; of a parabola, 5 ⁄ 7 of the height. If the vibrations are in the plane of the figure, then the radius of oscillation of a circle equals 3 ⁄ 4 of the diameter; of a rectangle, suspended at the vertex of one angle, 2 ⁄ 3 of the diag- onal. Center of Percussion.—For a body that moves without rotation, the resultant of all the forces acting on the body passes through the center of gravity. On the other hand, for a body that rotates about some fixed axis, the resultant of all the forces acting on it does not pass through the center of gravity of the body but through a point called the center of percus- Axis its diameter Axis at a distance Hollow Sphere Axis its diameter Thin Spherical Shell Ellipsoid Axis through center Paraboloid Axis through center k 0.6325r= k 2 2 5 r 2 = ka 2 2 ⁄ 5 r 2 += k 2 a 2 2 ⁄ 5 r 2 += k 0.6325 R 5 r 5 – R 3 r 3 – = k 2 2 R 5 r 5 –() 5 R 3 r 3 –() = k 0.8165r= k 2 2 3 r 2 = k b 2 c 2 + 5 = k 2 1 ⁄ 5 b 2 c 2 +()= k 0.5773r= k 2 1 3 r 2 = Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY 236 MOMENTS OF INERTIA sion. The center of percussion is useful in determining the position of the resultant in mechanics problems involving angular acceleration of bodies about a fixed axis. Finding the Center of Percussion when the Radius of Gyration and the Location of the Center of Gravity are Known: The center of percussion lies on a line drawn through the center of rotation and the center of gravity. The distance from the axis of rotation to the cen- ter of percussion may be calculated from the following formula in which q = distance from the axis of rotation to the center of percussion; k o = the radius of gyration of the body with respect to the axis of rotation; and r = the distance from the axis of rotation to the center of gravity of the body. Moment of Inertia An important property of areas and solid bodies is the moment of inertia. Standard for- mulas are derived by multiplying elementary particles of area or mass by the squares of their distances from reference axes. Moments of inertia, therefore, depend on the location of reference axes. Values are minimum when these axes pass through the centers of grav- ity. Three kinds of moments of inertia occur in engineering formulas: 1) Moments of inertia of plane area, I, in which the axis is in the plane of the area, are found in formulas for calculating deflections and stresses in beams. When dimensions are given in inches, the units of I are inches 4 . A table of formulas for calculating the I of com- mon areas can be found beginning on page 238. 2) Polar moments of inertia of plane areas, J, in which the axis is at right angles to the plane of the area, occur in formulas for the torsional strength of shafting. When dimensions are given in inches, the units of J are inches 4 . If moments of inertia, I, are known for a plane area with respect to both x and y axes, then the polar moment for the z axis may be calcu- lated using the equation, A table of formulas for calculating J for common areas can be found on page 249 in this section. When metric SI units are used, the formulas referred to in (1) and (2) above, are valid if the dimensions are given consistently in meters or millimeters. If meters are used, the units of I and J are in meters 4 ; if millimeters are used, these units are in millimeters 4 . 3) Polar moments of inertia of masses, J M * , appear in dynamics equations involving rota- tional motion. J M bears the same relationship to angular acceleration as mass does to linear acceleration. If units are in the foot-pound-second system, the units of J M are ft-lbs-sec 2 or slug-ft 2 . (1 slug = 1 pound second 2 per foot.) If units are in the inch-pound-second system, the units of J M are inch-lbs-sec 2 . If metric SI values are used, the units of J M are kilogram-meter squared. Formulas for calculating J M for various bodies are given beginning on page 250. If the polar moment of inertia J is known for the area of a body of constant cross section, J M may be calculated using the equation, where ρ is the density of the material, L the length of the part, and g the gravitational con- stant. If dimensions are in the foot-pound-second system, ρ is in lbs per ft 3 , L is in ft, g is * In some books the symbol I denotes the polar moment of inertia of masses; J M is used in this handbook to avoid confusion with moments of inertia of plane areas. qk o 2 r÷= J z I x I y += J M ρL g J= Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY MOMENTS OF INERTIA 237 32.16 ft per sec 2 , and J is in ft 4 . If dimensions are in the inch-pound-second system, ρ is in lbs per in 3 , L is in inches, g is 386 inches per sec 2 , and J is in inches 4 . Using metric SI units, the above formula becomes J M = ρLJ, where ρ = the density in kilograms/meter 3 , L = the length in meters, and J = the polar moment of inertia in meters 4 . The units of J M are kg · m 2 . Moment of Inertia of Built-up Sections.—The usual method of calculating the moment of inertia of a built-up section involves the calculations of the moment of inertia for each element of the section about its own neutral axis, and the transferring of this moment of inertia to the previously found neutral axis of the whole built-up section. A much simpler method that can be used in the case of any section which can be divided into rectangular elements bounded by lines parallel and perpendicular to the neutral axis is the so-called tabular method based upon the formula: I = b(h 1 3 - h 3 )/3 in which I = the moment of inertia about axis DE, Fig. 1, and b, h and h 1 are dimensions as given in the same illustration. Example:The method may be illustrated by applying it to the section shown in Fig. 2, and for simplicity of calculation shown “massed” in Fig. 3. The calculation may then be tabu- lated as shown in the accompanying table. The distance from the axis DE to the neutral axis xx (which will be designated as d) is found by dividing the sum of the geometrical moments by the area. The moment of inertia about the neutral axis is then found in the usual way by subtracting the area multiplied by d 2 from the moment of inertia about the axis DE. Tabulated Calculation of Moment of Inertia The distance d from DE, the axis at the base of the configuration, to the neutral axis xx is: The moment of inertia of the entire section with reference to the neutral axis xx is: Fig. 1. Fig. 2. Fig. 3. Section Breadth b Height h 1 Area b(h 1 - h) h 1 2 Moment h 1 3 I about axis DE A 1.500 0.125 0.187 0.016 0.012 0.002 0.001 B 0.531 0.625 0.266 0.391 0.100 0.244 0.043 C 0.219 1.500 0.191 2.250 0.203 3.375 0.228 Σ A = 0.644 ΣM = 0.315 ΣI DE = 0.272 bh 1 2 h 2 –() 2 bh 1 3 h 3 –() 3 d M A 0.315 0.644 0 . 4 9== = I N I DE Ad 2 –= 0.272 0.644 0.49 2 ×–= 0.117= Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY MOMENT OF INERTIA, SECTION MODULUS 239 A = bd y = d ⁄ 2 A = bd y = d A = bd y = 1 ⁄ 2 (d cos α + b sin α) A = bd − hk y = d ⁄ 2 Moments of Inertia, Section Moduli, and Radii of Gyration (Continued) Section A = area y = distance from axis to extreme fiber Moment of Inertia I Section Modulus Radius of Gyration Square and Rectangular Sections (Continued) Z I y = k I A = bd 3 12 bd 2 6 d 12 0.289d= bd 3 3 bd 2 3 d 3 0.577d= Abd= y bd b 2 d 2 + = b 3 d 3 6 b 2 d 2 +() b 2 d 2 6 b 2 d 2 + bd 6 b 2 d 2 +() 0.408 bd b 2 d 2 + = bd 12 d( 2 αcos 2 +b 2 αsin 2 ) bd 6 d 2 αcos 2 b 2 αsin 2 + d αcos b αsin+ ⎝⎠ ⎛⎞ × d 2 αcos 2 b 2 αsin 2 + 12 0.289 ×= d 2 αcos 2 b 2 αsin 2 + bd 3 hk 3 – 12 bd 3 hk 3 – 6d bd 3 hk 3 – 12 bd hk–() 0.289 bd 3 hk 3 – bd hk– = Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY MOMENT OF INERTIA, SECTION MODULUS240 Moments of Inertia, Section Moduli, and Radii of Gyration (Continued) Section Area of Section, A Distance from Neutral Axis to Extreme Fiber, y Moment of Inertia, I Section Modulus, Radius of Gyration, Triangular Sections 1 ⁄ 2 bd 2 ⁄ 3 d 1 ⁄ 2 bd d Polygon Sections ZIy⁄= kIA⁄= bd 3 36 bd 2 24 d 18 0.236d= bd 3 12 bd 2 12 d 6 0.408d= da b+() 2 da 2b+() 3 ab+() d 3 a 2 4ab b 2 ++() 36 ab+() d 2 a 2 4ab b 2 ++() 12 a 2b+() d 2 a 2 4ab b 2 ++() 18 ab+() 2 3d 2 30tan ° 2 0.866d 2 = d 2 A 12 d 2 12 30cos 2 °+() 430cos 2 ° 0.06d 4 = A 6 d 12 30cos 2 °+() 430cos 2 ° 0.12d 3 = d 2 12cos 2 30°+() 48cos 2 30° 0.264d= Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY MOMENT OF INERTIA, SECTION MODULUS 241 2d 2 tan 22 1 ⁄ 2 = 0.828d 2 Circular, Elliptical, and Circular Arc Sections Moments of Inertia, Section Moduli, and Radii of Gyration (Continued) Section Area of Section, A Distance from Neutral Axis to Extreme Fiber, y Moment of Inertia, I Section Modulus, Radius of Gyration, ZIy⁄= kIA⁄= 3d 2 30tan ° 2 0.866d 2 = d 230cos ° 0.577d= A 12 d 2 12 30cos 2 °+() 430cos 2 ° 0.06d 4 = A 6.9 d 12 30cos 2 °+() 430cos 2 ° 0.104d 3 = d 2 12cos 2 30°+() 48cos 2 30° 0.264d= d 2 A 12 d 2 12 22 1 ⁄ 2 ° cos 2 +() 422 1 ⁄ 2 ° cos 2 0.055d 4 = A 6 d 12cos 2 22 1 ⁄ 2 °+() 4cos 2 22 1 ⁄ 2 ° 0.109d 3 = d 2 12cos 2 22 1 ⁄ 2 °+() 48cos 2 22 1 ⁄ 2 ° 0.257d= πd 2 4 0.7854d 2 = d 2 πd 4 64 0.049d 4 = πd 3 32 0.098d 3 = d 4 πd 2 8 0.393d 2 = 3π 4–()d 6π 0.288d= 9π 2 64–()d 4 1152π 0.007d 4 = 9π 2 64–()d 3 192 3π 4–() 0.024d 3 = 9π 2 64–()d 2 12π 0.132d= π D 2 d 2 –() 4 0.7854 D 2 d 2 –()= D 2 π D 4 d 4 –() 64 0.049 D 4 d 4 –()= π D 4 d 4 –() 32D 0.098 D 4 d 4 – D = D 2 d 2 + 4 Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY MOMENT OF INERTIA, SECTION MODULUS242 πab = 3.1416ab a π(ab − cd) = 3.1416(ab − cd) a I–Sections bd − h(b − t) Moments of Inertia, Section Moduli, and Radii of Gyration (Continued) Section Area of Section, A Distance from Neutral Axis to Extreme Fiber, y Moment of Inertia, I Section Modulus, Radius of Gyration, ZIy⁄= kIA⁄= π R 2 r 2 –() 2 1.5708 R 2 r 2 –()= 4 R 3 r 3 –() 3π R 2 r 2 –() 0.424 R 3 r 3 – R 2 r 2 – = 0.1098 R 4 r 4 –() 0.283R 2 r 2 Rr–() Rr+ – I y I A πa 3 b 4 0.7854a 3 b= πa 2 b 4 0.7854a 2 b= a 2 π 4 a 3 bc 3 d–() 0.7854= a 3 bc 3 d–() π a 3 bc 3 d–() 4a 0.7854 a 3 bc 3 d– a = 1 ⁄ 2 a 3 bc 3 d– ab cd– b 2 2sb 3 ht 3 + 12 2sb 3 ht 3 + 6b 2sb 3 ht 3 + 12 bd h b t–()–[] Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY MOMENT OF INERTIA, SECTION MODULUS 243 dt + 2a(s + n) in which g = slope of flange = (h − l)/(b − t) = 10.0 ⁄ 6 for standard I-beams. bd − h(b − t) dt + 2a(s + n) in which g = slope of flange = (h − l)/(b − t) = 1 ⁄ 6 for standard I-beams. bs + ht + as d − [td 2 + s 2 (b − t) + s (a − t) (2d − s)]÷2A 1 ⁄ 3 [b(d − y) 3 + ay 3 − (b − t)(d − y − s) 3 − (a − t)(y − s) 3 ] Moments of Inertia, Section Moduli, and Radii of Gyration (Continued) Section Area of Section, A Distance from Neutral Axis to Extreme Fiber, y Moment of Inertia, I Section Modulus, Radius of Gyration, ZIy⁄= kIA⁄= d 2 1 ⁄ 12 bd 3 1 4g h 4 l 4 –()– 1 6d bd 3 1 4g h 4 l 4 –()– 1 ⁄ 12 bd 3 1 4g h 4 l 4 –()– dt 2as n+()+ d 2 bd 3 h 3 bt–()– 12 bd 3 h 3 bt–()– 6d bd 3 h 3 bt–()– 12 bd h b t–()–[] b 2 1 ⁄ 12 b 3 dh–()lt 3 g 4 b 4 t 4 –() + + 1 6b b 3 dh–()lt 3 g 4 b 4 t 4 –() + + I A I y I A Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY MOMENT OF INERTIA, SECTION MODULUS244 C–Sections dt + a(s + n) g = slope of flange for standard channels. dt + 2a(s + n) g = slope of flange g = slope of flange for standard channels. bd − h(b − t) Moments of Inertia, Section Moduli, and Radii of Gyration (Continued) Section Area of Section, A Distance from Neutral Axis to Extreme Fiber, y Moment of Inertia, I Section Modulus, Radius of Gyration, ZIy⁄= kIA⁄= d 2 1 ⁄ 12 bd 3 1 8g h 4 l 4 –()– hl– 2 bt–() = 1 ⁄ 6 = 1 6d bd 3 1 8g h 4 l 4 –()– 1 ⁄ 12 bd 3 1 8g h 4 l 4 –()– dt a s n+()+ bb 2 s ht 2 2 g 3 bt–() 2 b 2t+()× + + A÷ – hl– 2 bt–() = 1 ⁄ 3 2sb 3 lt 3 g 2 b 4 t 4 –()++ Ab y–() 2 – hl– 2 bt–() = 1 ⁄ 6 = I y I A d 2 bd 3 h 3 bt–()– 12 bd 3 h 3 bt–()– 6d bd 3 h 3 bt–()– 12 bd h b t–()–[] Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY [...]... Inertia 1. 00 1. 01 1.02 1. 03 1. 04 1. 05 1. 06 1. 07 1. 08 1. 09 1. 10 1. 11 1 .12 1. 13 1. 14 1. 15 1. 16 1. 17 1. 18 1. 19 1. 20 1. 21 1.22 1. 23 1. 24 1. 25 1. 26 1. 27 1. 28 1. 29 1. 30 1. 31 1.32 1. 33 1. 34 1. 35 1. 36 1. 37 1. 38 1. 39 1. 40 1. 41 1 .42 1. 43 1. 44 1. 45 1. 46 1. 47 1. 48 1. 49 0.0982 0 .10 11 0 .10 42 0 .10 73 0 .11 04 0 .11 36 0 .11 69 0 .12 03 0 .12 37 0 .12 71 0 .13 07 0 .13 43 0 .13 79 0 . 14 17 0 . 14 55 0 . 14 93 0 .15 32 0 .15 72 0 .16 13 0 .16 54 0 .16 96 0 .17 39... 3 9 14 688 399 819 8 40 83038 41 6 9220 42 56760 43 456 71 44 35968 45 276 64 4620775 4 715 315 4 811 298 49 08739 5007652 510 8053 5209956 5 313 376 5 41 8 329 55 248 28 10 3.5 10 4 10 4. 5 10 5 10 5.5 10 6 10 6.5 10 7 10 7.5 10 8 10 8.5 10 9 10 9.5 11 0 11 0.5 11 1 11 1.5 11 2 11 2.5 11 3 11 3.5 11 4 11 4. 5 11 5 11 5.5 11 6 11 6.5 11 7 11 7.5 11 8 11 8.5 11 9 11 9.5 12 0 12 0.5 12 1 12 1.5 12 2 12 2.5 12 3 12 3.5 12 4 12 4. 5 12 5 12 5.5 12 6 12 6.5 12 7 12 7.5 10 8 848 11 043 3... (mm) 10 7 10 8 10 9 11 0 11 1 11 2 11 3 11 4 11 5 11 6 11 7 11 8 11 9 12 0 12 1 12 2 12 3 12 4 12 5 12 6 12 7 12 8 13 0 13 2 13 5 13 8 14 0 14 3 14 7 15 0 15 5 16 0 16 5 17 0 17 5 18 0 18 5 19 0 19 5 200 210 220 230 240 250 260 270 280 290 300 … Moment of Inertia 10 2087 10 49 76 10 7 919 11 0 917 11 3969 11 7077 12 02 41 12 346 2 12 6 740 13 0075 13 346 8 13 6 919 14 043 0 14 40 00 14 7630 15 13 21 155072 15 8885 16 2760 16 6698 17 0699 17 47 63 18 3083 19 16 64 2050 31 219 006... 5. 34 5.35 5.36 5.37 5.38 5.39 5 .40 5. 41 5 .42 5 .43 5 .44 5 .45 5 .46 5 .47 5 .48 5 .49 12 .272 12 . 346 12 .42 0 12 .49 4 12 .569 12 . 644 12 . 719 12 .795 12 .870 12 . 947 13 .023 13 .10 0 13 .17 7 13 .2 54 13 .332 13 . 41 0 13 .48 8 13 .567 13 . 645 13 .725 13 .8 04 13 .8 84 13 .9 64 14 . 044 14 .12 5 14 .206 14 .288 14 .369 14 .4 51 14. 533 14 . 616 14 .699 14 .782 14 .866 14 . 949 15 .0 34 15 .11 8 15 .203 15 .288 15 .373 15 .45 9 15 . 545 15 .6 31 15. 718 15 .805 15 .892 15 .980... 11 043 3 11 20 34 11 3650 11 52 81 116 928 11 8590 12 0268 12 1962 12 3672 12 5398 12 713 9 12 8897 13 06 71 13 24 61 1 342 67 13 6089 13 7928 13 97 84 14 1656 14 3 545 14 545 0 14 7372 14 9 312 15 1268 15 32 41 1552 31 157238 15 9262 16 13 04 16 3363 16 544 0 16 75 34 16 9 646 17 1775 17 3923 17 6088 17 8270 18 04 71 182690 18 49 27 18 718 2 18 945 6 19 1 748 19 40 58 19 6386 19 87 34 2 011 00 20 348 4 5632890 5 742 530 5853762 5966602 60 810 66 619 716 9 6 3 14 927 643 4355 655 546 9... Inertia 4. 00 4. 01 4. 02 4. 03 4. 04 4.05 4. 06 4. 07 4. 08 4. 09 4 .10 4 .11 4 .12 4 .13 4 . 14 4 .15 4 .16 4 .17 4 .18 4 .19 4. 20 4. 21 4. 22 4. 23 4. 24 4.25 4. 26 4. 27 4. 28 4. 29 4. 30 4. 31 4. 32 4. 33 4. 34 4.35 4. 36 4. 37 4. 38 4. 39 4. 40 4. 41 4. 42 4. 43 4. 44 4 .45 4. 46 4. 47 4. 48 4. 49 6.2832 6.33 04 6.3779 6 .42 56 6 .47 36 6.5 218 6.5702 6. 618 9 6.6678 6. 716 9 6.7663 6. 815 9 6.8658 6. 915 9 6.9663 7. 016 9 7.0677 7 .11 88 7 .17 02 7.2 217 7.2736... 13 6 13 6.5 13 7 13 7.5 13 8 13 8.5 13 9 13 9.5 14 0 14 0.5 14 1 14 1.5 14 2 14 2.5 14 3 14 3.5 14 4 14 4. 5 14 5 14 5.5 14 6 14 6.5 14 7 14 7.5 14 8 14 8.5 14 9 14 9.5 15 0 … … … … 205887 208 310 210 7 51 213 211 215 690 218 188 220706 223 243 225799 2283 74 230970 2335 84 236 219 238873 2 41 5 47 244 2 41 246 9 54 249 688 25 244 2 255 216 258 010 260825 263660 266 516 269392 272288 275206 27 8 14 4 2 811 03 2 840 83 287083 29 010 5 29 3 14 8 296 213 299298 30 240 5... 308683 311 8 54 315 047 318 262 3 2 14 99 3 247 57 328037 3 313 40 … … … … 13 176795 13 383892 13 59 342 0 13 805399 14 019 848 14 236786 14 45 62 31 146 782 04 14 902723 15 129808 15 35 947 8 15 5 917 54 15 826653 16 06 41 9 8 16 3 044 06 16 547 298 16 792893 17 0 41 2 13 17 292276 17 54 610 4 17 802 715 18 06 213 1 18 3 243 72 18 58 945 8 18 857 41 0 19 128 248 19 4 019 93 19 678666 19 958288 20 240 878 2052 646 0 20 815 052 211 06677 2 14 013 56 216 9 910 9 219 99959 22303926 22 611 033... 6802 818 6929085 705 710 2 718 68 84 7 318 44 8 74 518 11 7586987 7723995 7862850 8003569 8 14 616 8 82906 64 843 70 74 8585 41 4 8735703 8887955 9 04 218 9 919 842 2 93566 71 9 516 953 9679286 9 843 686 10 010 172 10 178760 10 349 469 10 522 317 10 6973 21 108 744 98 11 053867 11 23 544 7 11 41 9 2 54 11 605307 11 793625 11 9 842 25 12 17 712 6 12 372 347 12 569905 12 769820 12 97 211 0 12 8 12 8.5 12 9 12 9.5 13 0 13 0.5 13 1 13 1.5 13 2 13 2.5 13 3 13 3.5 13 4 13 4. 5 13 5 13 5.5... 1. 40 87 1. 42 62 1. 44 38 1. 4 615 1. 47 94 1. 49 75 1. 515 6 0.78 54 0.8 012 0. 817 3 0.8336 0.85 01 0.8669 0.8 840 0.9 013 0. 918 8 0.9366 0.9 547 0.9730 0.9 915 1. 010 4 1. 0295 1. 048 9 1. 0685 1. 0885 1. 1087 1. 12 91 1 . 14 99 1. 1 710 1. 1923 1. 213 9 1. 2358 1. 25 81 1.2806 1. 30 34 1. 3265 1. 349 9 1. 3737 1. 3977 1. 42 21 1 .44 68 1. 4 717 1. 49 71 1.5227 1. 548 7 1. 5750 1. 6 016 1. 6286 1. 6559 1. 6836 1. 711 6 1. 7399 1. 7686 1. 7977 1. 82 71 1.8568 1. 8870 Copyright . 0.2752 0 .19 40 1. 91 0.68 41 0.6533 2. 41 1.3 742 1. 6559 1. 42 0.2 811 0 .19 96 1. 92 0.6 949 0.66 71 2 .42 1. 3 9 14 1. 6836 1. 43 0.28 71 0.2053 1. 93 0.7058 0.6 811 2 .43 1. 40 87 1. 711 6 1. 44 0.29 31 0. 211 1 1. 94 0. 716 8. 1. 213 9 1. 24 0 .18 72 0 .11 61 1. 74 0. 517 2 0 .45 00 2. 24 1. 10 34 1. 2358 1. 25 0 .19 17 0 .11 98 1. 75 0.5262 0 .46 04 2.25 1. 118 3 1. 25 81 1.26 0 .19 64 0 .12 37 1. 76 0.5352 0 .4 710 2.26 1. 1332 1. 2806 1. 27 0.2 011 0 .12 77. 10 41 6 .7 50 10 41 6 .7 41 6 .667 10 1 85858 .4 17 00 .17 260 14 646 67 11 266.7 51 110 54. 3 43 3.500 10 2 8 843 4.0 17 34. 00 270 16 40 250 12 150.0 52 11 717 .3 45 0.667 10 3 910 60.6 17 68 .17 280 18 29333 13 066.7 53 12 40 6.4