Explanation Center of mass: The masses of the systems were concentrated at a single point called the center of mass. Divide the system’s moment about the origin by the systems total mass to get the center of mass x ¯ and y ¯ Calculate center of mass of a thin plate covering a region in the xy plane is, x ¯ = M y M y ¯ = M x M Example: Consider the region bounded by the parabola y = x 2 and the line y = 4 . The density of the plate is δ . Calculation: The equation of parabola is y = x 2 . The equation of line is y = 4 . Sketch the region bounded by the parabola and the line as shown in Figure 1. Refer to Figure 1. The region is symmetrical about y axis. The mass is symmetric about the y axis and the center of mass lies on y axis. That is the center of mass along x axis x ¯ = 0 . Consider a vertical strip has center of mass ( x ˜ , y ˜ ) = ( x , x 2 + 4 2 ) . The length of strip is 4 − x 2 and width of strip is dx . Calculate area of the strip as shown below. d A = length × width Substitute 4 − x 2 for length and dx for width. d A = ( 4 − x 2 ) d x Calculate the mass of the strip as shown below. d m = δ d A Substitute ( 4 − x 2 ) d x for d A . d m = δ ( 4 − x 2 ) d x Calculate the mass of the plate as shown below. ∫ d m = ∫ − 2 2 δ ( 4 − x 2 ) d x M = δ ( 4 x − x 3 3 ) − 2 2 = δ ( 4 × 2 − 2 3 3 − ( 4 × ( − 2 ) − ( − 2 ) 3 3 ) ) = δ ( 8 − 8 3 + 8 − 8 3 ) = δ ( 16 − 16 3 ) = δ ( 48 − 16 3 ) = 32 δ 3 Calculate the moment of the strip about x axis as shown below

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 8GYR

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Chapter 6, Problem 7GYR

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