A hemisphere plate with diameter 6 ft is submerged vertically 3 ft below the sur

Question

A hemisphere plate with diameter 6 ft is submerged vertically 3 ft below the surface of the water. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Give your answer correct to the nearest whole number.) lb Point masses m_i are located on the x axis as shown. M_1 = 40, m_2 = 15, and m_3 = 25, Find the moment M of the system about the origin and the center of mass x bar. M = X bar = Calculate the moments M_x and M_y and the center of mass of a lamina with density rho = 8 and the given shape. M_x = M_y = (x bar, y bar) = (, ) Find exact coordinates of the centroid of the region bounded by the given curves. y = 3x + 4 y = x^2

Explanation / Answer

Solution:(2)

m1 = 40, m2 = 15, and m3 = 25

M = m1 + m2 + m3 = 40 + 15 + 25 = 80

Mo = (40)(-2) + 15(3) + 25(7) = -80 + 45 + 175 = 140

Cx = Mo/M = 140/80 = 7/4

Solution: (3)

This region may be described (presumably) by being bound between y = 2x/3,y = 0 and x = 3.

So, we have

m = (x, y) dA
= (x = 0 to 3) (y = 0 to 2x/3) 8 dy dx
= (x = 0 to 3) 8 * (2x/3 - 0) dx
= (x = 0 to 3) (16x/3) dx
= 16(x^2)/6 {for x = 0 to 3}
= 16*9/6
= 24

My = x (x, y) dA
= (x = 0 to 3) (y = 0 to 2x/3) 8x dy dx
= (x = 0 to 3) (8x)(2x/3-0) dx
= (x = 0 to 3) (16(x^2)/3) dx
= (16/3)(x^3)/3 {for x = 0 to 3}
= (16/9)(3^3)
= 48

Mx = y (x, y) dA
= (x = 0 to 3) (y = 0 to 2x/3) 8y dy dx
= (x = 0 to 3) 4y^2 {for y = 0 to 2x/3} dx
= (x = 0 to 3) 16(x^2)/9 dx
= (16/9)(x^3)/3 {for x = 0 to 3}
= 16

So, the center of mass (My / m, Mx / m) equals
(48/24, 16/24) = (2, 2/3) = (x, y)

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