A vertical plate is partially submerged in water and has the indicated shape. The height is 3m, the width at the top is 4m, and the width at the bottom is 8m. Label the shape (trapezoid) appropriately. Express the hydrostatic force against one side of the plate as an integral and evaluate it, if the water level is 1 m from the top of the plate.

1.98 × 10^4 N

Explanation:

Form similar triangle ADE and ABC

a/x= 2/3, a=2/3x

Width of the strip w= 2(4+a) = 8+2a

W= 8 +2 (2/3x)= 8+4/3x

Area of the strip = w Δx

(8 +4/3x) Δx

Pressure on the strip p= pgx= 10^3 ×9.81x= 9810x

But,

Force= Pressure × area= 9810x × (8+4/3x)Δx

Adding the forces and taking lim n to infinity

F total= lim n--> infinity E 9810x × (8+4/3x)Δx

Ftotal= Integral 2,0 9810x × (8+4/3x)Δx

F total= 9810 integral 2, 0 (8+4/3x)dx

= 9810(8+x^2/2 + 4/3x^3/3)2,0

=9810(4x^2 + 4/9x^3)

=9810(4x2^2 + 4/9×2^3-0)

=9810(16 + 32/9)

Hydrostatic force as an integral

Ft= 19.18 ×10^4N

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