Fluid Mechanics: Topic 4.1 - Hydrostatic force on a plane surface

Summary

This lesson explores the derivation of the hydrostatic force equation on a plane surface, essential for assessing the structural integrity of a water-filled fish tank. It explains calculating the resultant force on the bottom and side walls, considering atmospheric pressure and water depth. The video also introduces the concept of the centroid and center of pressure, illustrating how to integrate these to find the resultant force on an arbitrarily shaped and oriented wall, crucial for fluid mechanics problem-solving.

Takeaways

• 📚 The lesson aims to derive the equation for calculating the hydrostatic force on a plane surface submerged in a fluid.
• 🏞️ A rectangular fish tank filled with water to a depth 'd' is used as an example to explain the concept of hydrostatic force.
• 🌡️ The atmospheric pressure 'Pa' surrounds the tank and contributes to the force exerted on the tank walls.
• 🔽 The bottom wall of the tank experiences a resultant force due to the difference in pressure between the inside and outside of the tank.
• 📉 The atmospheric pressure terms cancel out in the calculation of the resultant force on the bottom wall, leaving a force proportional to the specific weight of the water 'gamma' and the depth 'd'.
• 🔼 The side walls of the tank experience varying pressures due to the linear increase in pressure with depth inside the tank.
• 📏 A coordinate system is established to analyze the pressure forces on an arbitrary section of the side wall at an angle 'theta' to the free surface.
• 📐 The net force due to fluid pressure at a small area 'dA' on the side wall is calculated using trigonometry, considering the depth 'h' as 'y sin(theta)'.
• ∫ The resultant force on the entire wall is found by integrating the net force over all points of the wall, leading to an expression involving the centroid of the wall.
• 📌 The center of pressure 'CP' is identified as the point where the resultant force 'FR' acts, equating to the pressure at the centroid 'PC' times the area.
• 🔧 The equations derived are applicable for any flat surface where the liquid's free surface and the tank's side are exposed to the same pressure, such as atmospheric pressure.
• 🛠️ If the tank is pressurized to a gage pressure 'P0', the resultant force equation must be adjusted to account for this additional pressure.

Q & A

What is the purpose of deriving the equation for calculating the magnitude of the hydrostatic force on a plane surface?

The purpose is to determine whether the fish tank walls can support the water pressure without breaking, which is essential for structural integrity and safety.

Why is it necessary to know the resultant force along the bottom wall of a fish tank?

Knowing the resultant force along the bottom wall helps in understanding the total force exerted on the walls, which is crucial for assessing the tank's ability to withstand the water's pressure.

What is the atmospheric pressure denoted by in the script?

The atmospheric pressure is denoted by 'Pa' in the script.

How does the pressure force on the underside of the bottom wall of the tank relate to the area of the bottom wall?

The pressure force on the underside of the bottom wall is equal to the atmospheric pressure 'Pa' times the area of the bottom wall, and it points upward.

What is the formula for the resultant force acting along the bottom wall of the tank?

The resultant force is calculated by subtracting the downward force (Pa + gamma * d) times the area from the upward force Pa times the area, resulting in -gamma * d * (area of the bottom wall).

What is the significance of the atmospheric pressure terms canceling out in the calculation of the resultant force on the bottom wall?

The cancellation simplifies the equation, leaving only the component of the force due to the water's weight, which is gamma times d times the area, pointing downward.

How does the pressure along the interior of the side walls of the tank change with depth?

The pressure along the interior of the side walls increases linearly with depth, being the atmospheric pressure plus the specific weight of the water times the depth at any point.

What is the role of the centroid in calculating the resultant force on a wall section?

The centroid is the geometric center of the wall and is used to determine the location of the center of pressure, which is where the resultant force acts, simplifying the calculation of the force on the wall section.

What is the expression for the resultant force FR on the entire wall in terms of the centroid?

The resultant force FR is equal to gamma times sin(theta) times yC (the y-coordinate of the centroid) times the area A of the wall.

How can the resultant force equation be rewritten using the pressure at the centroid?

The resultant force equation can be rewritten as the pressure at the centroid PC times the area A, where PC is gamma hC, the pressure at the centroid's depth.

What modification would be needed to the resultant force equation if the tank is pressurized to a gage pressure of P0?

The equation would be modified to (gamma sin(theta) yC plus P0) times the area A, which can also be expressed as (gamma hC plus P0) times A.