10-133* Determine the second moments I, Iy, and Iy forthe composite area shown in Fig. P10-133.2 in.5 in3 in.5 in.7 in.10 in.Fig. P10-133

Answered: 10-133* Determine the second moments I,…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

10-133* Determine the second moments I, Iy, and Iy for
the composite area shown in Fig. P10-133.
2 in.
5 in
3 in.
5 in.
7 in.
10 in.
Fig. P10-133

Expert Solution

Step 1

Refer to the above fig.

$H e r e w e h a v e T r i a n g l e h a v i n g a r e a A_{1} R e c \tan g l e h a v i n g a r e a A_{2} C i r c l e h a v i n g a r e a A_{3}$

Step 2

$N o w a c c o r d i n g t o g i v e n c o n d i t i o n F o r t r i a n g l e A_{1} = \frac{b x h}{2} A_{1} = \frac{10 x 10}{2} A_{1} = 50 i n^{2} N o w m o m e n t a b o u t x - a x i s i s g i v e n b y I_{x_{1}} = \frac{b h^{3}}{12} I_{x_{1}} = \frac{10 x 10^{3}}{12} I_{x_{1}} = 833.33 i n^{4} M o m e n t a b o u t y - a x i s i s g i v e n b y I_{y_{1}} = \frac{h b^{3}}{12} I_{y_{1}} = \frac{10 x 10^{3}}{12} I_{y_{1}} = 833.33 i n^{4} M i x e d s e c o n d m o m e n t o f i n e r t i a i s g i v e n b y {(I_{x y})}_{1} = \frac{b^{2} h^{2}}{12} {(I_{x y})}_{1} = \frac{10^{2} x 10^{2}}{12} {(I_{x y})}_{1} = 416.6 i n^{4}$

Step 3

$N o w a c c o r d i n g t o g i v e n c o n d i t i o n A_{2} = b x h A_{2} = 7 x 10 A_{2} = 70 i n^{2} N o w m o m e n t o f r e c \tan g l e a b o u t x - a x i s i s g i v e n b y I_{x_{2}} = \frac{h b^{3}}{3} I_{x_{2}} = \frac{7 x 10^{3}}{3} I_{x_{2}} = 2333.33 i n^{4} M o m e n t o f r e c \tan g l e a b o u t y - a x i s i s g i v e n b y I_{y_{2}} = \frac{h b^{3}}{3} I_{y_{2}} = \frac{10 x 7^{3}}{3} I_{y_{2}} = 1143.33 i n^{4} M i x e d s e c o n d m o m e n t o f i n e r t i a r e c \tan g l e g i v e n b y {(I_{x y})}_{2} = {({I_{x y}}_{C})}_{2} + d_{x_{2}} . d_{y 2} . A_{2} {(I_{x y})}_{2} = 0 + (5) (- 3.5) (70) {(I_{x y})}_{2} = - 1225 i n^{4}$