Explanation Show the element parallel to the y axis (shaded region) as in Figure (1). Show the element parallel to the x axis (shaded region) as in Figure (2). Using Figure (1), Write the mass of element parallel to the y axis. d m = ρ d V = ρ 0 x y ( t y d x ) = ρ 0 t x y 2 d x (I) Here, the constant is ρ 0 , thickness of the plate is t , differential thickness of plate is dx , and positions are x and y respectively. Express the parabolic equation. x 2 + y 2 = r 2 y 2 = r 2 − x 2 Substitute the value of y 2 in Equation (I). d m = ρ 0 t x ( r 2 − x 2 ) d x = ρ 0 t ( r 2 x − x 3 ) d x (II) Write the mass of element parallel to the x axis using Figure (2). d m = ρ d V = ρ 0 x y ( t x d y ) = ρ 0 t x 2 y d y Substitute r 2 − y 2 for x 2 . d m = ρ 0 t ( r 2 − y 2 ) y d y = ρ 0 t ( r 2 y − y 3 ) d y Compute the mass of the differential element. m = ∫ m d m (III) Compute the centroid of the differential element along x axis using the formula. x ˜ = x Compute the centroid of the differential element along y axis using the formula. y ˜ = y Determine the location of the centre of gravity of the plate along x axis ( x ¯ ) using the relation from Figure (2). x ¯ = ∫ m x ˜ d m ∫ m d m (IV) Determine the location of the centre of gravity of the plate along y axis ( y ¯ ) using the relation. y ¯ = ∫ m y ˜ d m ∫ m d m (V) Conclusion: Apply the limits from 0 to r in Equation (III) and substitute the value of dm from Equation (II). m = ∫ 0 r ρ 0 t ( r 2 x − x 3 ) d x = [ ρ 0 t ( r 2 x 2 2 − x 4 4 ) ] 0 r = 1 4 ρ 0 r 4 t Thus, the mass of the plate, m is 1 4 ρ 0 r 4 t _

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 9.1, Problem 37P

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The mass of the plate, m.

The centroid (x¯,y¯) of the mass of the plate.

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Chapter 9.1, Problem 36P

Chapter 9.1, Problem 38P

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Chapter 9 Solutions

Engineering Mechanics: Statics

Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid of the shaded area. Prob....Ch. 9.1 - Locate the center of mass x of the straight rod if...Ch. 9.1 - Locate the centroid of the homogeneous solid...Ch. 9.1 - Locate the centroid z of the homogeneous solid...Ch. 9.1 - Locate the center of mass of the homogeneous rod...Ch. 9.1 - Prob. 2P Ch. 9.1 - Prob. 3P Ch. 9.1 - Prob. 4P

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The mass of the plate, m . The centroid ( x ¯ , y ¯ ) of the mass of the plate.

The mass of the plate, m. The centroid (x¯,y¯) of the mass of the plate.

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Chapter 9 Solutions

The mass of the plate, m.

The centroid (x¯,y¯) of the mass of the plate.