Question

A hemispherical plate with diameter 6 ft is submerged vertically 1 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.) 2δ 3 Correct: Your answer is correct. 0 dy ≈ lb

Answer 1

Answer:

F = 7476 N

Step-by-step explanation:

given,

diameter of hemispherical plate = 6 ft

height of submergence = 1 ft

the weight density of water =  62.5 lb/ft³

Assuming that hemispherical plate is residing on x and y axis.

bottom of plate is on x-axis and left side of the plate touches y-axis

now, plate is defined by the upper half of the circle

(x - 3)² + (y-0)² = 3²

  y² = 9 - (x - 3)²

  y = √(9 - (x - 3)²)

hydro static pressure on one  side of plate.

$F = \int \rho g x w(x)dx$

$F = \int_0^3 62.5\times 9.8 x \times \sqrt{9-(x-3)^2}dx$

$F = 612.5 \int_0^3 x \times \sqrt{9-(x-3)^2}dx$

on solving the above equation

$F = 612.5(27\dfrac{\pi}{4}-9)$

F = 7476 N