A vertical plate is submerged in water and has the indicated shape.
A triangle is submerged in water pointed up. The top most vertex of the triangle touches the surface of the water. The base of the triangle is 6 m and is parallel to the surface of the water. The height of the triangle is 15 m.
Express the hydrostatic force (in N) against one side of the plate as an integral (let the positive direction be upwards) and evaluate it. (Use 9.8 m/s2 for the acceleration due to gravity. Recall that the weight density of water is 1,000 kg/m3.)
Answers
The hydrostatic force is given by the integral of the force acting on
small horizontal strips from the base to the top of the triangle.
Response:
The hydrostatic force one one side of the plate is 4.41 MNHow can the hydrostatic force acting on the plate be calculated?The given parameters are;
Base of the triangle = 6 m
Height of the triangle = 15 m
Acceleration due to gravity, g ≈ 9.8 m/s²
Weight density of water, ρ ≈ 1,000 kg/m³
Required:
Hydrostatic force on one side of the plate
Solution:
Considering a thickness, dx, having area, dA = y·dx
The pressure on the strip = ρ·g·x
Force on the strip, dF = ρ·g·x·y·dx
Using similar triangles, we have;
[tex]\dfrac{x}{15} = \dfrac{y}{6}[/tex]
Which gives;
[tex]y = \dfrac{x}{15} \times 6 = \mathbf{ \dfrac{2}{5} \cdot x}[/tex]
Therefore;
[tex]dF = \rho \cdot g \cdot x \cdot \dfrac{2}{5} \cdot x \cdot dx = \mathbf{\rho \cdot g \cdot x^2 \cdot \dfrac{2}{5} \cdot dx}[/tex]
Which gives;
[tex]F =\displaystyle \int\limits^{15}_0 { \rho \cdot g \cdot x^2 \cdot \dfrac{2}{5} } \, dx = \mathbf{\left[\rho \cdot g \cdot x^3 \cdot \dfrac{2}{3\times 5} \right]^{15}_0}[/tex]
Therefore;
[tex]F = \rho \cdot g \cdot \left[x^3 \cdot \dfrac{2}{3\times 5} \right]^{15}_0 = \mathbf{ \rho \cdot g \times 15^3 \cdot \dfrac{2}{3\times 5}} = 450 \cdot \rho \cdot g[/tex]
Which gives;
F = 450 m³ × 1,000 kg/m³ × 9.8 m/s² = 4,410,000 N = 4.41 MN
The hydrostatic force one one side of the plate, F = 4.41 MNLearn more about hydrostatic force here:
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5.4
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Answers
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i hope this helps! have a wonderful rest of ur day :)
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Step-by-step explanation:
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Hope it helps have a great day :)