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 answer:
Answer: 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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Answer: (a) 9.00 Mega Newtons or 9.00 * 10^6 N
(b) 17.1 m
Explanation: The length of wall under the surface can be given by
The average pressure on the surface of the wall is the pressure at the centeroid of the equilateral triangular block which can be then be calculated by multiplying it with the Plate Area which will provide us with the Resultant force.
Noting from the Bernoulli equation that
From the second image attached the distance of the pressure center from the free surface of the water along the surface of the wall is given by:
Substituting the values gives us the the distance of the surface to be equal to = 17.1 m
