To solve this problem, we need to express the hydrostatic force on a vertical submerged circular plate in the form of an integral and evaluate it.

Step 1: Hydrostatic Force Concept

The hydrostatic force on a submerged surface is given by the integral:

$F = \int \rho g y \, dA$

Where:

• $\rho$ is the density of water ($1000 \, \text{kg/m}^3$),
• $g$ is the acceleration due to gravity ($9.8 \, \text{m/s}^2$),
• $y$ is the depth at which the differential area $dA$ is located (measured from the surface of the water),
• $dA$ is the differential area of the surface.

Step 2: Set up the Geometry

The plate is circular with radius $7 \, \text{m}$, submerged vertically. If we center the plate at the origin (with $y = 0$ at the water surface), the equation of the circle is:

$x^2 + z^2 = 7^2$

For vertical integration, it's easier to work with horizontal slices. The width of a horizontal slice at a particular depth $y$ can be found using the equation for the circle, which gives the radius of the slice at that depth:

$x = \sqrt{7^2 - y^2}$

This is the radius of the circular slice at height $y$, so the differential area $dA$ for each slice is:

$dA = 2x \, dy = 2\sqrt{7^2 - y^2} \, dy$

Step 3: Force Integral

Now, substitute into the force equation:

$F = \int_{-7}^{7} \rho g y \cdot 2\sqrt{7^2 - y^2} \, dy$

Here, $y$ is the vertical depth, and the limits of integration are from $-7$ to $7$, corresponding to the vertical extent of the plate.

Step 4: Simplify and Evaluate the Integral

$F = \int_{-7}^{7} 1000 \cdot 9.8 \cdot y \cdot 2\sqrt{49 - y^2} \, dy$

Now, let's evaluate the integral.

We can use symmetry of the problem. Since the plate is symmetric about the water surface, the force on the upper half is equal in magnitude to the force on the lower half but opposite in direction. Hence, we only need to compute the integral for the upper half (from 0 to 7) and then double the result.

So, the integral becomes:

$F = 2 \times 1000 \times 9.8 \int_{0}^{7} y \sqrt{49 - y^2} \, dy$

Step 5: Perform the Integration

The integral can be solved by using a substitution or by recognizing it as a standard form. Let's calculate the integral and multiply by the constants:

$\int y \sqrt{49 - y^2} \, dy$

This is a standard integral. The result of the integral is:

$\frac{-1}{3} (49 - y^2)^{3/2}$

Evaluating from 0 to 7:

$\left[ \frac{-1}{3} (49 - y^2)^{3/2} \right]_{0}^{7}$

At $y = 7$, the expression becomes $0$, and at $y = 0$, it evaluates to:

$\frac{-1}{3} (49)^{3/2} = \frac{-1}{3} (343) = -114.33$

Now, multiply the result by $2 \times 1000 \times 9.8$:

$F = 2 \times 1000 \times 9.8 \times 114.33 = 2242526.4 \, \text{N}$

Final Answer:

So, the total hydrostatic force is approximately:

$F \approx 2,242,526 \, \text{N}$

Let me know if you'd like further details or have any questions!

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Here are some related questions you may find interesting:

1. How do you calculate the hydrostatic force on a rectangular submerged surface?
2. How does the depth of submersion affect the hydrostatic force on an object?
3. What is the role of fluid density in calculating hydrostatic force?
4. Can the same method be applied to a non-circular surface submerged in water?
5. How does the orientation of a submerged plate influence the force calculation?

Tip: When calculating forces on submerged surfaces, always remember the importance of symmetry; it can often simplify the problem significantly!