Explanation Given: The solution for the Exercise 11(a) For Square with vertices ( 5 , 0 ) , ( a + 5 , 0 ) , ( 5 , a ) , ( a + 5 , a ) and density ρ = k . The center of the mass is: ( x ¯ , y ¯ ) = ( a 2 + 5 , a 2 ) 11(b) For Square with vertices ( 5 , 0 ) , ( a + 5 , 0 ) , ( 5 , a ) , ( a + 5 , a ) and density ρ = k y . The center of the mass is: ( x ¯ , y ¯ ) = ( a 2 + 5 , 2 a 3 ) . 11(c) For Square with vertices ( 5 , 0 ) , ( a + 5 , 0 ) , ( 5 , a ) , ( a + 5 , a ) and density ρ = k x . The center of the mass is: ( x ¯ , y ¯ ) = ( 2 ( a 2 + 15 a + 75 ) 3 ( a + 10 ) , a 2 ) 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. ( x ¯ , y ¯ ) = ( M y m , M x m ) The integral of, ∫ x n d x = x n + 1 n + 1 + c . Calculation: For a square with vertices ( 0 , 0 ) , ( a , 0 ) , ( 0 , a ) , ( a , a ) and density ρ = k . The mass of the lamina is: m = ∫ 0 a ∫ 0 a k d y d x = ∫ 0 a [ k y ] 0 a d x = [ k a x ] 0 a = k a 2 To find the center of the mass, find the moment of inertia about both axis

11th Edition

ISBN: 9781337275576

Author: Larson, Ron, Edwards, Bruce

Publisher: Cengage Learning,

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Chapter 14.4, Problem 12E

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To calculate: The conjecture about the change in center of mass when a lamina of a constant density is translated c units horizontally or d units vertically.

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Chapter 14.4, Problem 11E

Chapter 14.4, Problem 13E

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The conjecture about the change in center of mass when a lamina of a constant density is translated c units horizontally or d units vertically.

Solution Summary: The author calculates the conjecture about the change in center of mass when a constant density is translated c units horizontally or d units vertically. This is not necessarily true for variable densities.

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To calculate: The conjecture about the change in center of mass when a lamina of a constant density is translated c units horizontally or d units vertically.

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Chapter 14 Solutions