# How Does a Sphere Act as a Valve in Fluid Dynamics?

• Feynmanfan
In summary: This will involve taking into account the pressure at all points on the surface and using the outward normal to determine the direction of the force.
Feynmanfan
Fluid mechanics problem.

In the picture below we have a sphere (consider it massless) acting as a valve in a water tank. At the bottom, there's a hole (which magnitude is determined by the angle theta.

I am asked to calculate the total vertical force acting on the sphere. And after that, when theta is (3Pi/4) what must the height h be so that the water doesn't flow through the hole.

I can calculate the pressure on any point of the sphere but how do I calculate this vertical force? Is it archimedes i have to use?

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Where is the picture?

sorry. I forgot the picture

do I have to integrate over the sphere? just give me a hint please.

Feynmanfan said:
Fluid mechanics problem.

In the picture below we have a sphere (consider it massless) acting as a valve in a water tank. At the bottom, there's a hole (which magnitude is determined by the angle theta.

I am asked to calculate the total vertical force acting on the sphere. And after that, when theta is (3Pi/4) what must the height h be so that the water doesn't flow through the hole.

I can calculate the pressure on any point of the sphere but how do I calculate this vertical force? Is it archimedes i have to use?

Archimedes principle is equivalent to a pressure calculation over the surface of a submerged object, assuming pressure at any depth is uniform and increases with depth due only to the weight of the fluid. A floating object displaces its own weight of fluid because pressure at any depth exceeds the pressure at a lesser depth by the weight of a column of fluid of cross-sectional area A divided by that area. Under any other conditions, Archimedes principle is not going to hold.

If you know the pressure at all points on the surface of your sphere, you should be able to calculate the vertical force. It should have the form

$$\overrightarrow F = -\oint P \widehat n dA}$$

if $$\widehat n$$ is the outward normal

The net force on the upper hemisphere will be vertically downward. The net force on the lower hemisphere will be vertically upward. If there were no drain, the lower pressure would exceed the upper pressure and the upward force would exceed the downward force by the weight of the displaced water. With the drain, the upward force will be reduced. If you don't add atmospheric pressure to the pressure at all levels, you can take the pressure over the region spanned by the drain opening as zero. This region results in lower upward force, and when it is just the right size the downward force and the upward forces will cancel. At that point the ball will still be displacing a lot of water, so what does that say about Archimedes principle?

Feynmanfan said:
do I have to integrate over the sphere?

Yes

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## 1. What is a sphere in a water tank?

A sphere in a water tank refers to a spherical object or ball that is placed inside a tank filled with water.

## 2. What is the purpose of a sphere in a water tank?

The purpose of a sphere in a water tank is to simulate the behavior of a floating object in water and to study the effects of buoyancy and pressure on the object.

## 3. How is a sphere in a water tank used in scientific experiments?

A sphere in a water tank is used in various scientific experiments to understand the principles of fluid mechanics, such as Archimedes' principle and Bernoulli's principle.

## 4. What factors affect the behavior of a sphere in a water tank?

The behavior of a sphere in a water tank is affected by several factors, including the density and shape of the sphere, the density and temperature of the water, and the depth and size of the tank.

## 5. What are the potential applications of studying a sphere in a water tank?

Studying a sphere in a water tank can have various applications in fields such as engineering, oceanography, and meteorology. It can help in the design and testing of ships, submarines, and other water vessels, as well as in understanding ocean currents and weather patterns.

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