# Calculate Hydrostatic Force Exerted on Plane Submerged in Water

In summary, the conversation is discussing finding the hydrostatic force exerted on a plane submerged vertically in water. This can be calculated using the basic definitions of distance from surface (d), density (p), and pressure (P). The conversation also mentions setting up the problem correctly and the use of the Riemann sum or integral to find the approximate or exact force on the plate.
I need to find the hydrostatic force exerted on a plane submerged vertically in water. I attached a diagram of the problem.

Here are the basic definitions:
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d=distance from surface, p=density, P=pressure

$$p=\frac{m}{V}$$

$$P=pgd=\delta d$$

$$F=mg=pgAd$$

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The area of the ith strip is $$A_i=6\Delta y$$ so the pressure exerted on the ith strip is $$\delta d_i=pgd_i=pg(6-y_i^*)$$

The hydrostatic force on the ith strip is $$F_i=\delta_iA_i=6pg(6-y_i)\Deltay$$

The approximate force along the entire surface is therefore:

$$F_{net}=\lim_{n-\infty}\Sigma_{i=1}^n6pg(6-y_i)\Delta y$$

$$=6pg\int_0^4(6-y)dy$$

Am I setting this up correctly?

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It looks like you are doing it correctly from first principles, but I think this line should be (not sure on notation but this is how I saw a similar summation in a math book)

$$F_{net}=\lim_{\Delta y \rightarrow 0} \sum_{y=0} ^{y=4} 6pg(6-y_i)\Delta y$$

I was writing out the limit of the Riemann sum. There are $$n$$ subdivisions and $$\Delta y=\frac{4-0}{n}$$. So I think what you wrote was equivalent to the Riemann sum.

I was writing out the limit of the Riemann sum. There are $$n$$ subdivisions and $$\Delta y=\frac{4-0}{n}$$. So I think what you wrote was equivalent to the Riemann sum.

It probably is, I was never taught the Riemann Sum, but you are correct though.

You don't need to write out the Riemann sum; that just makes things unnecessarily complicated. I find that going directly to Fnet$=6pg\int_0^4(6-y)dy$ is much easier and more intuitive. (BTW, that integral gives the exact force on the plate, not the approximate force.)

## 1. How do you calculate the hydrostatic force exerted on a plane submerged in water?

To calculate the hydrostatic force, you will need to know the density of water, the depth at which the plane is submerged, and the area of the plane that is in contact with the water. The formula for calculating hydrostatic force is F = ρghA, where ρ is the density of water, g is the acceleration due to gravity, h is the depth of the submerged plane, and A is the area of the plane in contact with the water.

## 2. What is the density of water?

The density of water is 1000 kg/m³ at 4°C. This value may vary slightly depending on the temperature and salinity of the water, but for most calculations, 1000 kg/m³ can be used as an accurate estimate.

## 3. How does the depth of the submerged plane affect the hydrostatic force?

The depth of the submerged plane is directly proportional to the hydrostatic force exerted on it. This means that as the depth increases, the hydrostatic force also increases. This is because the pressure exerted by the water increases with depth, resulting in a higher force being exerted on the plane.

## 4. Can the shape of the plane affect the hydrostatic force?

Yes, the shape of the plane can affect the hydrostatic force. The formula for calculating hydrostatic force assumes that the plane is a flat surface, but if the plane has a curved or angled shape, the force may be different. In this case, the area of the plane in contact with the water would need to be adjusted accordingly.

## 5. How can the hydrostatic force be used in real-world applications?

The hydrostatic force is an important concept in many engineering and scientific fields. It is used in the design of dams, ships, and other structures that are submerged in water. It is also used in calculating the buoyant force on objects, which is important in determining whether an object will float or sink in water. Understanding hydrostatic force is crucial for ensuring the safety and stability of structures and objects in water environments.

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