(a) Explanation Given information: The density of the plate is δ . The region cut from the first quadrant by the circle x 2 + y 2 = 9 . Calculation: The radius of the circle is r = 3 . Provide the Equation of circle as shown below. x 2 + y 2 = 9 y 2 = 9 − x 2 y = 9 − x 2 (1) Sketch the region cut from the first quadrant by the circle x 2 + y 2 = 9 as shown in Figure 1. Refer to Figure 1. The plate is symmetric about the line x = y . Hence, the center of mass x ¯ = y ¯ . Calculate the area of the quarter circle of radius 3 as shown below. A = π r 2 4 Substitute 3 for r . A = π × 3 2 4 = 9 π 4 Consider a vertical strip has center of mass ( x ˜ , y ˜ ) = ( x , 9 − x 2 2 ) The length of the strip is 9 − x 2 and the width of strip is dx . Calculate area of the strip as shown below. d A = length × width Substitute 9 − x 2 for length and dx for width. d A = 9 − x 2 d x Calculate the mass of the strip as shown below. d m = δ d A Substitute 9 − x 2 d x for d A . d m = δ 9 − x 2 d x Calculate the mass of the plate as shown below. M = δ × A Substitute 9 π 4 for A . M = δ × 9 π 4 = 9 π δ 4 Calculate the moment of the strip about x axis as shown below (b) To determine Calculate the center of mass of a thin plate.

University Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780321999580

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 6.6, Problem 10E

(a)

To determine

Calculate the center of mass of a thin plate.

(b)

To determine

Calculate the center of mass of a thin plate.

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Chapter 6.6, Problem 9E

Chapter 6.6, Problem 11E

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