University:
Michigan State University
Course:
ERG 160 | Success in Science, Technology, Engineering, and Mathematics
Academic year:

2024
Views:

59

Pages:

3
Author:

Adowj7b

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 24. x = r 2 − y2 First note: JP = Ix + I y r I x = ∫ y 2dA = 2∫ r ∫ 0 r 2 − y2 2 y 2dxdy r = 2∫ r y 2 r 2 − y 2 dy 2 Let Then Thus y = r sin θ dy = r cosθ dθ ; y = r: θ = π 2 ; y= r : 2 θ = π 6 π I x = 2∫ π2 r 2 sin 2 θ ( r cosθ )( r cosθ dθ ) 6 Now sin 2θ = 2sin θ cosθ Ix = thus 1 2 sin 2θ 4 1 4 π2 2 r ∫ π sin 2θ dθ 2 6 1  θ sin 4θ  = r4  −  2 2 8  π 2 π 6 = 1 4π π 1 3 + × r  −  2  4 12 8 2  = r4  π 3  +  4 3 8  r I y = ∫ x 2dA = 2∫ r ∫0 r 2 − y2 2 = sin 2 θ cos 2 θ = x 2dxdy 3 2 r 2 2 ∫ r r − y 2 dy 3 2 ( ) continued COSMOS: Complete Online Solutions Manual Organization System Let Then Now Thus y = r sin θ Iy = 2 π2 3 3 ∫ π r cos θ 3 6 ( ) ( r cosθ dθ ) ( ) cos 4 θ = cos 2 θ 1 − sin 2 θ = cos 2 θ − Iy = 1 2 sin 2θ 4 2 4 π2  2 1  r ∫ π  cos θ − sin 2 2θ  dθ 3 4 6   π 2  θ sin 2θ  1  θ sin 4θ   2 = r 4  + −  −  3  2 4  4 2 8  π 6 Then =  π 3 3  π  π π r 4  − ×  −  + × − × + × ×  3  2 4 2   6 2 2 4 6 4 4 2   = r4  π 3 3   −  4  3 8  JP = = r4  π 3  r4  π 3 3   + +  −  43 8  4  3 8  r4 8π − 3 3 48 ( ) J P = 0.415r 4 Now r A = 2∫ r xdy 2 r = ∫ r r 2 − y 2 dy 2 continued COSMOS: Complete Online Solutions Manual Organization System Let y = r sin θ π A = 2∫π2 ( r cosθ )( r cosθ dθ ) 6 π 2 θ sin 2θ  2 = 2r  +  4 π 2 6 π π 1 3 = r 2  − − ×  2  2 6 2 = Have kP = = r2 4π − 3 3 12 ( ) JP A r4 8π − 3 3 48 r2 4π − 3 3 12 ( ( ) ) k P = 0.822 r

COSMOS Chapter 9 Solution 24

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