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Theodore R. (Theodore Rudolph) Running.

Empirical formulas online

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The volume is by (7)

^•i'i(i49-oo4)=3i.2 cu. units.

Centroids
Let the coordinates of the centroid of an area be represented
by X and y. Then from the calculus

I xydx
x =



I ydx

- I y'^dx
ydx



y=



The integral in the numerator of the value of x may be
represented by the area bounded by the curve Y = xy, the :r-axis
and the two ordinates x = a and x = b. The original area is
bounded by the curve whose ordinates are represented by y,
the a;-axis and the two ordinates x = a and x = b. The integral
in the numerator of the value of y may be represented by the
area bounded by the curve Y=y^, the a:-axis and the two
ordinates x = a and x = b.

For a volume generated by revolving a given area about the
rc-axis

TT I y^xdx



TT I y'^dx



NUMERICAL INTEGRATION

When the volume is irregular



127



x =



Axdx



X



Adx



The process of finding the coordinates of the centroid of the
area in Fig. 28 is shown in the table:



y

xy

3,2

yH






.2


•4


.6


.8


1.0


I.O


I-S


2.2


2.7


2.6


2-3


0.00


0.30


0.88


1.62


2.08


2.30


1. 00


2.25


4.84


7.29


6.76


5-29


0.000


0.450


1.936


4-374


5-408


5.290



1.2

2.1
2.52
4.41
5.292



The area under the curve Y = xy is

i^[o.oo+i.2o+i.76+6.48+4.i6+9.2o+2.52] = 1.688;
- 1.688



x =



2.58



= .654.



The area under the curve Y = ^y^ is
^[i.oo+9.cx>+9.68 + 29.i6 + 13.524-21.16+4.41] = 2.931

-_2-93i



2.58



1. 136.



As was pointed out before, large changes in the ordinates
must be avoided.

For the volume generated by revolving the area about
the x-sods

-_7ri^[o.ooo+ 1. 800+3.872 + 17.496+ 10. 816+ 21. 160+ 5. 292]
7r3^[i.oo+9.oo+9.68+29.i6+i3.52+2i.i6+4.4i]

60.436



87-93



.687.



128



EBiPIRICAL FORMULAS



Moments of Ineriia

The expression for the moment of inertia of an area about
the 3^-axis is

Iy= j x^ydx.

About the ac-axis

/*= I xfdy.

When the equation of the curve is known these integrals
can be calculated at once, but when this is not the case approxi-
mate methods must be resorted to.

I. The process of finding the approximate values of these
integrals is shown in the table below. The values of x and y
are taken from Fig. 28.



X

y

x'^y

\y'






.2


.4


.6


.8


1.0


I.O


1-5


2.2


2.7


2.6


2.3


O.CXXJ


0.060


0.335


0.972


1.664


2.300


0.333


I. 125


3 549


6.561


5.859


4.056



I. 2
2.1
3.024

3 087



If the values of x'^y be plotted to x we will have a curve
under which the area represents the moment of inertia of the
area in Fig. 28 about the 3/-axis.

Dividing this by the area found before, there results for the
radius of gyration about the y-axis



Plotting iy^ to X and finding the area under the curve so
determined

7^ = 4.6136,
and

i?/ = 1.788.



NUMERICAL INTEGRATION



129



2. The form of a quarter section of a hollow pillar, Fig. 31,
is given by the following table. Find the moment of inertia
of the section about the axes of x and y.



y
.5
















n


~~->














A


~^


\




^


K




N






\








.3


N






\






\






\








\




\












\














\






•i










\






















\







Fig. 31.





X


F


xW '


y


X


y^X




00


050


. 00000


00


100


. 00000




05


055


.00014


05


108


.00027




10


068


.00068


10


116


.00116




15


078


.00175


IS .


120


.00270




20


096


.00384


20


125


.00500




25


116


.00725


25


130


.00812




30


148


.01332


30


133


.01197




35


200


.02450


35


140


.01715




40


300


. 04800


40


150


.02400




45


215


■04354


45


215


•04354




50


000


. 00000


50


000


.00000



In the above table X stands for the width of the area parallel
to the X-axis and Y for the width parallel to the ;y-axis. The
area is 0.066.

The moment of inertia about the y-3Lxis is



X



x^Ydx = .ooj;^6;

i?-:52M = o.iiis.

.060



130 EMPIRICAL FORMULAS

The moment of inertia about the nc-axis is

I y^Xdy = .oo6ig;
Jo

^'^ = ^' = -^38,

where R stands for the radius of gyration.

The values of the above integrals were computed by for-
mula (7).



APPENDIX



If a chart could be constructed with all the different forms
of curves together with their equations which may arise in
representing different sets of data it would be a comparatively-
simple matter to select from the curves so constructed the one
best suited for any particular set. Useful as such a chart would
be its construction is clearly out of the question. The most
that can be done of such a nature is to draw a number of curves
represented by each one of the simpler equations.

A word of caution is, however, necessary here. A particular
curve may seem to the eye to be the one best suited for a given
set of data, and yet, when the test is applied, it may be found
to be a very poor fit. It is of some aid, nevertheless, to have
before the eye a few of the curves represented by a given formula.

The purpose of the following figures is to illustrate the
changes in the form of curves produced by shght changes in
the constants. Figs. I, II, III, and IV show changes produced
by the addition of terms. Figs. V to XIX changes in form
produced by changes in the values of the constants, and Fig.
XX the changes in form brought about by varying both the
values of the constants and the number of terms.

A discussion of all the figures is unnecessary. A few words
in regard to one will suffice. Formula XIV, for example,
y = a-\-bx'', an equation which can be made to express fairly
well the quantity of water flowing in many streams if x
stands for mean depth and y for the discharge per second,
represents a family of triply infinite number of curves. Fixing
the values of b and c and varying the value of a does not
change the form of the curve, but only moves it up or down

131



132



EMPIRICAL FORMULAS



along the >'-axis. Keeping the values of a and b constant and
varying the value of c, the formula will represent an infinite
number of curves all cutting the ^'-axis in the same point. In
the same way, keeping the value of a and c constant and vary-
ing the value of b, an infinite number of curves is obtained,
all of which cut the ^'-axis in a fixed point. In Fig. XIV the
quantity a is constant and equal to unity, while b and c vary.

To one trained in the theory of curves the illustrations are,
of course, of no essential value, but to one not so trained they
may be of considerable help.

The text should be consulted in connection with the curves
in any figure. The figures are designated to correspond to the
formulas discussed in the first five chapters.




(1) y=l-.lx

(2) 2/=l-.lx+.01x2

(3) 2/=l-.lx+.01a;2_.001xS

(4) j/=l-.lx+.01x2-.001ar34-

.OOOlx*

(5) 2/=l-.lx+.01x2-.001x'+

.OOOlx^-.OOOOlx^

(6) 2/=-l-.lx+.01a:2-.001a;'+

.OOOlx^-.OOOOlx^+.OOOOOlx^

See formula I, page 13



Fig. I.



APPENDIX



133



V
1.4


ll




















































1.2


4-

























'


4-




























\'2)










OX




%


^


^
























"/


/






















0.4
0.2





' /


























/
























f


















































-0.2
-0.4


i














































































-0.8




























-1,


Jul




























(1) y=l-l/x

(2) 2/=l-l/x+l/x2

(3) y=l-l/x+l/x^-l/x^

(4) y=l-l/x+l/x'^-l/x^+l/x*

(5) y=l-l/x+l/x^-l/x^+

l/x*-l/x^

(6) y=l-l/x+l/x^-l/x^+

1/x^-l/x^+l/x^

See formula II, page 22



10 a u X



Fig. II.



3

2.8


y-












/


/


j






















/


/




























/


/










2.2

2

1.8

1.6












(1)/




//
























•y


i




















/




//




















/


/


/;


/




















/J


^


/^..^






N,










1.4


— .




y


C^


^






^


X


sA


^








1.2

a

0.8
0.6
0.4
0.2







^


^
















^






























\


























s\


























N




























































8


>


I


J


3


K


J


) 1


1


i 1


d -x



(1) -=l-.la;

V

(2) -=l-.lx+.01x2

L3) -=l-.lx+.01x2-.001a;3
(4) -=l-.lx+.01x2-.001.3f

y

.000 Ix*



(5) -=l-.lx+.01x2-.00U3+
V

.0001x4-. OOOOlx^

(6) -=l-.lx+.01x2-.001x3+

y

.OOOlx*-. 00001x6+. OOOOOlx®
See formula III, page 25



Fig. in.



134



EldPIRICAL FORMULAS



1




























w




























1JI
















































































/.


























A


,y^






"^


^


=^


^


^




J2L


^


^


^


















■^


^





























>


s^


:^


^


























N


s\




























^




























Jl




























//j








•<w


1

1


{


J


*


5


e




8




\


1 u


I h


r-t



(1) y«=l-.lx

(2) i/«=l-.lx+.01x2

(3) v»=l-.li+.01x*-.001x»

(4) y»-l-.lx+.01x«-.001x«+

.000 Ix*

(5) 1/2=1-. Ix+.01x2-.001x»+

.0001x*-.00001x'

(G) 2/2=1-. Ix+.01x2_.001a:'' +

.0001x*-.00001x*+.000001x«

See formula IV, page 25



Fig. IV.















/










/


^






(1) l/=(.5)^

(2) l/=(.6)*

(3) 2/= (.7)*

(4) l/=(.8)*

(5) y=(.9)*

(6) y=(.95)*

(7) y=.99)*

(8) j/=(1.01)*

(9) i/=(1.05)*

(10) 2/ = (l.l)*

(11) 2/=(1.2)*

See formula, "^

a














y












/




















/










/




















/


^^








/






















/








/






















/


/






f














^










/






/












^


^










/






/










^


^














/




y








^


















/ ^


y




^^
























A-r











(S)










v











r~~"






—I






Sl




















\>


c


^^




■ -






•^


















^V




\




"^




-^








^














\


S:


^




-^


■^








■^


—-


.







>^, page






1


^


^


^






— ■



















!




«


s


e


3


e


«


1


1


I \.


8 1


J 14


U





Fig. V.



APPENDIX



135



V




























1 8






(li.


■i)^


-^^^^


■::z^


^=




,,^






_^








/


K




(4)^


^


;::::;




^^










1.4




y/


/
^


^
^


^


[6)^


^-■










^






y.


\^






(7)


9)


-—


nj











1




^>^


===:
























R


^^







i::


.(11)
















0.8
0.6




■"






'^




:>N


■^


(12)























\




"S.




\






^^




-02












n


\


■n








^-v




















\


\


























\


N


\
























\


— s
S


























\








%


I


\


£


«




t


s


1


) 1


I 1


I »



(1) 2/ = 2-(.5)*

(2) |/=2-(.6)*

(3) y=2-(.7)*

(4) j/=2-(.8)*

(5) y-2-(.85)*

(6) y=2-(.9)*^

(7) 2/ = 2-(.95)*

(8) 2/=2-(.97)*

(9) 2/=2-(.99)*

(10) 2/=2-(1.01)*

(11) y=2-(1.03)^

(12) y=2-(1.05)*
(I3)y=2-(1.07)*
(14) j/=2-(1.08)*

See formula VI, page 28



Fig. VI.







































^




^


'^^




^







^










J


c-


y


^^


^




^^


-^












/


//


>


y


^


^


















//


v


/


^i.








^


r1












//




y




(5)^


- ^



















//


^






(6)


,-^



















^

X








^(7);

-(-8^









































\^


\^






"-^




















s^


v^


^^


U^





__




"










































(1) log2/=l-.5(.5)*

(2) logy=l-.5(.6)^

(3) logy=l-.5(.7)^

(4) log2/=l-.5(.8)^

(5) logy=l-.5(.9)*

(6) log2/=l-.5(.95)*

(7) logy=l-.5(.98)*

(8) log2/=l-.5(1.02)'^

(9) logy=l-.5(l.l)*

(10) logy=l-.5(1.2)3J

(11) log2/=l-.5(1.3)a^

(12) log2/=l-.5(1.5)a'

(13) log2/=l-.5(2)a?
base=10

See formula VII, page 32



Fig. VII.



136



EMPIRICAL FORMULAS



V







































(1)














____




u


i




^


"




J3),


■^^
















1 1






-^




"^














^-


N


\


^


Sw.


^


^^










1


r








\


•^


s.




^


^




-^














^


N


N


X


\


^


^


















\




N




V


N,




1
















\




\


N
























\






N
























\








•OS






















\




























\





(1) i/=2-.01x-(.6)*

(2) v=2-.03a;-(.5)*

(3) v = 2-.05i-(.5)*

(4) v-2-.08x-(.5)*

(5) v=2-.lx-(.5)*

(6) i/=2-.12x-(.5)*

(7) j/=2-.15x-(.5)*

(8) i/=2-.2x -(.5)*

See formula VIII, page



15 6 7 8 9

Fig. VIII



V


























































1






















1






























































\




























\






















/






\\






















//






\


u>


















/


/




f


\


\


















/


/






L


\\


















^


/




1


-ar




^




^


,








^










(St,


^




^






—'




^


\








































I


i


I


i


»


r


}


) 1


Q 1


1 1


2 »



(1) j/=10-81-.36x + .03x«

(2) j/= 10-54 -.24x + . 02x2

(3) y=10-27-.12x + .01x2

(4) y= 10-135 -.0&r + .005x«

(5) j/=10 - 135 + .06x- 005x2

(6) y = 10 - 54 + -24x-.02x2

See formula IX, page 37



Fig. IX.



APPENDIX



137



V.


























































\




























\






















i'




1.6


\


















(1)^


^

[(2)^






1.4


\










^


^


?^


^




(3)










V-







""^


















\




















/ i








1


\
















y


/










N


5s














•/


/












^


^










^


.y
























: — ■




(6)
























5


i






6


r


i


9 1


1


I J


^ *



(1) y=(1.01)* (1.05)(l-2)^

(2) j/=(1.01)* (1.05)<1-16)^

(3) y=(1.01)* (1.05)(1-15)^

(4) 2/= (.5)== (2) (1.24)^

(5) j/=(.5)^ (2) (1-23)^

(6) j/=(.5)* (2) (1-2)''

See formula X, page 37



Fig. X.































f


\
























/


\


\






















/




\




















1


/


\


■\


\^


















1






\,


\


<<-
















V




\


\


^->


\


V




.











L


>,




\^.


\


X


\


^










/-


\


■^<j




\






.^^


^




.^^






r


N






-^






-~


— .




^







1




r




"~—


' —




















4 5 6 7

Fig. XI.



9 10 U 12 «



(1) y=



(2) V-



(3) y-



(4) y=



(5) y



.2-


.lx+.05x2




X


2—


.lx+.07x2




X


.2-


.lx+.la;2




X


.2-


.la;+.2x2




X



.2-.lx+.4i2
See formula XI, page 38



138



EMPIRICAL FORMULAS



» -




























































y


y
















,






f


/


















" 1




/










^^












' \




/




K^










.








'\




^''




J2-


















Jl




::;;-




-JiL


















1


f \


v^




jw__











__








]\








^






__






































ol—


1 i


f


i




\ (


\




) I


1


i> 1


1 1


2 »



(1) y=5x-*

(2) i/=5x-2

(3) V-5X-1

(4) |/=5x - l

(5) j/=5x - 2

(6) y^Sx - *

See formula XII, page 42



Fig. XII.



y

t

18

te

14
It

























































































































































.
























-fc


























^


::^


















u


1








u=


^






















I




y


^














__^












w


\


/


/




^


^























\


^


















lO,








"






y


\










.


^




















/






^


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(S)





































1




























































,








( 1






]


^IG


X


II]


1


1


I 1


2 1


3 1


1 1


\



(l)y=l+Iogx+.llog2x;

y=— 1.5 (min.) when log x= —5

(2)|/=l+logx+.011og2x;

2/= —24 (min.) when log x= —50

(3) y=l+.21ogx+.31og2x

(4) y=l— log x+log^ X

(5) v=l-logx+.51og*x

See formula XIII, page 44



APPENDIX



139





































































































^


^^


















^^








■^











^


^






=^












^


^






- -






(i)




















^






><^>























X


N




^
























N^




























\






I i




i


.


> (




r


3


) 1


I


1 1


2 ■»



(1) y=l+.008xl-7

(2) y=l+.007a;l-6

(3) y=l+.006xl-5

(4) 2/= 1-. 002x2

(5) y=l-.003a;2.1

(6) 2/= 1-. 004x2.2

See formula XIV, page 45



Fig. XIV.

























































^




























\
























^




\
























\




\


















^\


^




\


\


\
















\


k


\




\


\


V














^.


^


K


N

N,


\


\


N
















^


\

~~^,






:x.


\

s.
























b:


-<v.






^










i


i


4


f


(


]




1


^ 1


, 1


r-t



(1) y=(2.0) 10 - 01^^

(2) 2/= (1.6) 10- •02x1-''

(3) 2/= (1.2) 10 - 03i^-^

(4) 2/= (1.0) 10 - 04x^-^^

(5) 2/= (0.8) 10 - 05a;l-24

(6) 2/= (0.6) 10- •06x1-12

See formula XV, page 49



Fig. XV.



140



EMPIRICAL FORMULAS

































































^




5


















^


^
%


:^













^


^






















w


























7\


n1


s
























\,


K


\


/''




















V


\


V


^
























^


j^


" - ^


^







■-^




















"^




___;








«^


i


a


\


8


t




6


t


Y 1


1


1


e X



(1) (v-2) (xf.S) 1

(2) (v-2) (x+.75) = -1.5

(3) (v-2) (x+l) = -2

(4) (1/+.1) (x+4)=8.2

(5) (1/+.1) (x+3) = 6.3

(6) (1/+.1) (x+2)=4.2

See formula XVI, page 53



Fig. XVI.



V
























~


















































































I 1




























1




^




















y


^






\






^




-^










y








\


^.1








3-






^^




















.^




-f2h;




















1


"^







J4)






„.




i













































J5L
















•8


(


























1


\


S




!




} \




9


) )


1


1 1


» s>



24

(1) y=\ 10a;+24

-12

(2) j/=J lCa;-24

2

(3) 2/=i 10a;+2

2_

(4) 2/=i35 10^

■5

(5) 2/=-10a: + l

See formula XVIo, page 56



Fig. XVIa.



APPENDIX



141



v.


C


1













































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^




















^


,^




















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^


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1.4


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(1) y=.5e-01a:+e.05a;

(2) y=2e-05a;_.5e.lx

(3) y=2.25e-05a;_.75e.Ix

(4) y=1.8e-01a;_.3e.li

(5) 2/=1.92e - l^-.42e - 01a;

(6) y=2e - 05a:_e-.01a;

(7) y=4.2e - 2a;-3.5e - 25a;

(8) y=4.5e - 2a;-4.1e - 25a:

(9) y=.25e - 01aJ-.]Ce - 152

(10) y=e- lx-i,u-.2x

(11) y=.27e - 01^-.77e - 25a;

(12) y=e - la;-2e - 25a;

See formula XVII, page 58



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(1) j/=c.01a;(i.5 cos .lx-.5 sin .Ix)

(2) j/=e~'2a^(1.5cos.5x— .5sin.5x)

(3) 2/=e - la:(.6cos .lx+.8sin .Ix)

(4) j/=e-1^(.2 cos .3x-.l sin .3x)

(5) j/=e-02a;(.4cos.l6x+.17sin.l6x)

(6) j/-=.5e~'^^ sin X

See formula XVIII, page 61



Fig. XVIII



142



EMPIRICAL FORMULAS



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Fig. XIX.



(1) i/=2x-l-i-2

(2) v=3i-5-2.2x-8

(3) y-2.3x-8-2x.85

(4) y=.lx-l+.5x-2

(5) t/=.33x-.0012x3 ^

(6) i/=.25x-5+.05x-8

See formula XIX, page 65




(1) y=15xl-5(.4)'^

(2) y=3x2(.5)* ^

(3) j/=3x-2(i.5)*

(4) y=-.5xl-5(.75)*

See formula XlXa, page 72



Fig. XlXa.



APPENDIX



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Fig. XX.



(1) 2/=166.25— 14.5 cos x-2.75 cos 2x— 10 sin x

(2) y=167.83-20 cos x-4.33 cos 2x+5.5 coa 3x-13.28 sin x-17.32 sin 2x

(3) i/=167.62-17.5 cos x-2.75 cos 2x+3 cos 3x-1.38 cos 4x-12.42 sin x-18 sin 2x-

2.42 sin 3x

(4) y= 167.08-17.22 cos x-3. 5 cos 2x+5.5 cos 3x-0.83 cos 4x-2.78 cos 5x+

0.75 cos 6x— 12.14 sin X— 19.05 sin 2x— sin 3x— 1.73 sin 4x+ 1.14 sin 5x



See formula XX, page 74



INDEX



Approximation, Accuracy of, 114, 118,

119, 121
Area, 11, 114, 121, 124, 125, 126, 128

Graphical Determination of, 121
Arithmetical Series, 11, 13, 15, 19, 22,

2S» 27, 28, 32, 33, 37, 38, 44, 58, 61, 104

Centroids, 11, 126, 127
Check, 46, 92

Common Difference, 11, 16
Compound Interest Law, 27
Constants, 11, 92, 94, 131, 132

Graphical Determination of, 13, 15,
21, 23, 24, 25, 28, 31, 32, 36, 40,
44, 47, 48, 51, 55, 60, 64, 65, 67 ,70

Changes in, 131
Curves, 10, 11, 114, 121, 122, 131, 132

Damped Vibration, 65
Derivative, 95

First, 108

Second, 108
Differences, 11, 12, 16, 33, loi, 115,
n6

First Order, 12, 18

Higher Order, 12, 16, 18, 19, 22, loi
Differential Equation, 68, 71, in
Differentiation of Tabulated Functions,

100, 108

Elastic Limit, 14

Empirical Formulas, 9, n, 68, 90, 100
Errors of Observation, 11, 68
Fourier Series, 74


1 2 3 4 5 7

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