Explanation Given: The lamina is bounded by the functions, x = y + 2 , x = y 2 Formula used: The moments about x - axis and y - axis of a planar lamina are, M y = ρ ∫ a b [ f ( y ) + g ( y ) 2 ] [ f ( y ) − g ( y ) ] d y And, M x = ρ ∫ a b y [ f ( y ) − g ( y ) ] d y And, the Center of mass ( x ¯ , y ¯ ) is, x ¯ = M y m And, y ¯ = M x m And, mass of lamina is, m = ρ ∫ a b [ f ( y ) − g ( y ) ] d y Calculation: The graph of the provided equations x = y + 2 , x = y 2 is as follows: Here, the lower bound is y = − 1 and upper bound is y = 2 as the intersection points of graph are y = − 1 and y = 2 . Now, moment about y -axis of the planar lamina is, M y = ρ ∫ a b [ f ( y ) + g ( y ) 2 ] [ f ( y ) − g ( y ) ] d y Here, f ( y ) = y + 2 , g ( y ) = y 2 , a = − 1 , b = 2 . Substitute the above values to get, M y = ρ ∫ − 1 2 [ y + 2 + y 2 2 ] [ y + 2 − y 2 ] d y = ρ 2 ∫ − 1 2 ( y + 2 ) 2 − y 4 d y = ρ 2 ∫ − 1 2 ( y 2 + 4 + 4 y − y 4 ) d y = ρ 2 [ y 3 3 + 4 y + 2 y 2 − y 5 5 ] − 1 2 On further calculation, M y = ρ 2 [ ( 2 ) 3 3 + 4 ( 2 ) + 2 ( 2 ) 2 − ( 2 ) 5 5 + 1 3 + 4 − 2 − 1 5 ] = ρ 2 [ 3 + 18 − 33 5 ] = 36 ρ 5 Now, moment about x -axis of the planar lamina is, M x = ρ ∫ a b y [ f ( y ) − g ( y ) ] d y Here, f ( y ) = y + 2 , g ( y ) = y 2 , a = − 1 , b = 2

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Chapter 7.6, Problem 28E

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To calculate: The Mx, My and (x¯,y¯) for the lamina is bounded by the functions, x=y+2, x=y2 of uniform density ρ.

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Chapter 7.6, Problem 27E

Chapter 7.6, Problem 29E

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The M x , M y and ( x ¯ , y ¯ ) for the lamina is bounded by the functions, x = y + 2 , x = y 2 of uniform density ρ .

Solution Summary: The author explains that the lamina is bounded by the functions, x = y+2, x=y2 of uniform density.

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To calculate: The Mx, My and (x¯,y¯) for the lamina is bounded by the functions, x=y+2, x=y2 of uniform density ρ.

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Chapter 7 Solutions