(a) Explanation Given: The lamina R : Triangle with vertices ( 0 , 0 ) , ( a , 0 ) , ( a , a ) and density ρ = k . Formula used: Mass of planar lamina is: m = ∬ R ρ ( x , y ) d A Moment of mass of variable density planar lamina is: M x = ∬ R ( y ) p ( x , y ) d A M y = ∬ R ( x ) p ( x , y ) d A The center of mass is: ( x ¯ , y ¯ ) = ( M y m , M x m ) The integral of ∫ x n d x = x n + 1 n + 1 + c . Calculation: The mass of the lamina is: m = ∫ 0 a ∫ 0 y k d x d y = ∫ 0 a [ k y ] 0 y d y = ∫ 0 a k y d y On further simplification, m = [ k y 2 2 ] 0 a = k a 2 2 To find the center of the mass, find the moment of inertia about both axis (b) To determine To calculate: The mass and center of mass of the lamina corresponding to region, R : Triangle with vertices ( 0 , 0 ) , ( a , 0 ) , ( a , a ) and density ρ = k y . (c) To determine To calculate: The mass and center of mass of the lamina corresponding to region, R : Triangle with vertices ( 0 , 0 ) , ( a , 0 ) , ( a , a ) and density ρ = k x .

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Chapter 14.4, Problem 7E

(a)

To determine

To calculate: The mass and center of mass of the lamina corresponding to region, R: Triangle with vertices (0,0), (a,0), (a,a) and density ρ=k.

(b)

To determine

To calculate: The mass and center of mass of the lamina corresponding to region, R: Triangle with vertices (0,0), (a,0), (a,a) and density ρ=ky.

(c)

To determine

To calculate: The mass and center of mass of the lamina corresponding to region, R: Triangle with vertices (0,0), (a,0), (a,a) and density ρ=kx.

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Chapter 14.4, Problem 6E

Chapter 14.4, Problem 8E

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Chapter 14 Solutions

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The mass and center of mass of the lamina corresponding to region, R : Triangle with vertices ( 0 , 0 ) , ( a , 0 ) , ( a , a ) and density ρ = k .

Solution Summary: The author explains the formula used to calculate the mass and center of mass of the lamina corresponding to region, R.

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To calculate: The mass and center of mass of the lamina corresponding to region, R: Triangle with vertices (0,0), (a,0), (a,a) and density ρ=k.

To calculate: The mass and center of mass of the lamina corresponding to region, R: Triangle with vertices (0,0), (a,0), (a,a) and density ρ=ky.

To calculate: The mass and center of mass of the lamina corresponding to region, R: Triangle with vertices (0,0), (a,0), (a,a) and density ρ=kx.

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Chapter 14 Solutions