Explanation Given: R be the rectangle [ − a , a ] × [ b , − b ] with uniform density and total mass M. Formula used: Mass density δ = total mass/total area I x = ∬ R y 2 δ ( x , y ) d A I y = ∬ R x 2 δ ( x , y ) d A I o = I x + I y M = ∬ R δ ( x , y ) d A R x = I x M Where R x is the radius of gyration about x axis, I x is the moment of inertia about x axis and I o is the moment of inertia about origin. Calculation: Part a) Mass density δ = total mass/total area = M 4 ∫ 0 a ∫ 0 b 1 d y d x = M 4 ∫ 0 a [ y ] 0 b d x = M 4 b ∫ 0 a 1 d x = M 4 b [ x ] 0 a = M 4 a b For part b) I x = ∬ R y 2 δ ( x , y ) d A = 4 ∫ 0 a ∫ 0 b y 2 . M 4 a b d y d x = 4 M 4 a b ∫ 0 a [ y 3 3 ] 0 b d x = M 3 a b ∫ 0 a [ y 3 ] 0 b d x = M b 3 3 a b ∫ 0 a 1 d x = M b 3 3 a b [ x ] 0 a = M a b 3 3 a b = M b 2 3 I y = ∬ R x 2 δ ( x , y ) d A = 4 ∫ 0 a ∫ 0 b x 2 ...

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ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Question

Chapter 16.5, Problem 31E

To determine

To find: a)The mass density of rectangle R [ $[- a, a] \times [b, - b]$ ]

b)The moment of inertia about x axis $I_{x}$ and about origin $I_{o}$

c)The radius of gyration about x axis.

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Chapter 16.5, Problem 30E

Chapter 16.5, Problem 32E

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Chapter 16 Solutions

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a)The mass density of rectangle R [ [ − a , a ] × [ b , − b ] ] b)The moment of inertia about x axis I x and about origin I o c)The radius of gyration about x axis.

Solution Summary: The author calculates the mass density of the rectangle R, the moment of inertia about x axis, and the radius of gyration.

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To find: a)The mass density of rectangle R [ $[- a, a] \times [b, - b]$ ]

b)The moment of inertia about x axis $I_{x}$ and about origin $I_{o}$

c)The radius of gyration about x axis.

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Chapter 16 Solutions

a)The mass density of rectangle R [ [ − a , a ] × [ b , − b ] ] b)The moment of inertia about x axis I x and about origin I o c)The radius of gyration about x axis.

Solution Summary: The author calculates the mass density of the rectangle R, the moment of inertia about x axis, and the radius of gyration.

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To find: a)The mass density of rectangle R [[−a,a]×[b,−b]] b)The moment of inertia about x axis Ix and about origin Io c)The radius of gyration about x axis.

Want to see the full answer?

Chapter 16 Solutions

To find: a)The mass density of rectangle R [ $[- a, a] \times [b, - b]$ ]

b)The moment of inertia about x axis $I_{x}$ and about origin $I_{o}$

c)The radius of gyration about x axis.