(a) Explanation Given information: The region is bounded by the curve y 2 = 4 a x and by the line x = a . Here, a is a positive constant. The density at the point ( x , y ) is proportional to x . Calculation: The plate is symmetric about x -axis. Therefore, the centroid y ¯ = 0 . Consider a small vertical strip with the center of mass of ( x ˜ , y ˜ ) = ( x , 0 ) , length 4 a x , width d x , and area 4 a x d x . Mass of the small vertical strip is d m = δ d A . For some proportionality constant k and d A = 4 a x d x . d m = k x ⋅ 4 a x d x Find the moment of the strip about y -axis using the relation: M y = ∫ x ˜ d m Substitute 4 k x a x d x for d m . M y = ∫ 4 k x 2 a x d x = 4 k a ∫ 0 a x 5 / 2 d x = 4 k a [ x 7 / 2 7 / 2 ] 0 a = 4 k a 1 / 2 ( 2 7 a 7 / 2 ) = 8 k a 4 7 Find the moment of the strip using the relation: M = ∫ d m Substitute 4 k x a x d x for d m (b) To determine Obtain the center of mass of a thin plate if the density at the point ( x , y ) is proportional to y .

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 6, Problem 14AAE

(a)

To determine

Obtain the center of mass of a thin plate if the density at the point (x,y) is proportional to x.

(b)

To determine

Obtain the center of mass of a thin plate if the density at the point (x,y) is proportional to y.

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University Calculus: Early Transcendentals (4th Edition)

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Obtain the center of mass of a thin plate if the density at the point ( x , y ) is proportional to x .