A hemispherical plate with diameter 6 ft is submerged vertically 2 ft below the surface of the water. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Round your answer to the nearest whole number. Recall that the weight density of water is 62.5 lb/ft3.)

Hydrostatic Force = 14952.35N

Step-by-step explanation:

From the question, we are given;

Diameter of hemispherical plate = 6 ft

Height of submergence = 2ft

Weight density of water = 62.5 lb/ft³

Now, if we assume that the hemispherical plate is residing on x and y axis, then bottom of the plate is on x-axis while the left side of the plate touches the y-axis

Now, the plate is defined by the upper half of the circle as;

(x - 3) ² + (y-0) ² = 3²

(x - 3) ² + y² = 9

Thus, y² = 9 - (x - 3) ²

y = √ (9 - (x - 3) ²)

To solve this, hydro static force on one side of plate is given as;

F = ∫ ρgd•xw (x) δx =

2∫ρgx√ (9 - (x - 3) ²) δx at boundary of 3 and 0

F = 62.5•9.8•2∫x√ (9 - (x - 3) ²) δx at boundary of 3 and 0

F = 1225[ (27π/4) - 9]

F = 1225 x 12.206 = 14952.35N