Calculating Volume: Integrating Pressure & Area

[werson tan]

Homework Statement

why the formula of volume is given by
integral of P and dA , the integral of P and dA would yield Force , right ?

Homework Equations

The Attempt at a Solution

 

[SteamKing]

Homework Statement

why the formula of volume is given by
integral of P and dA , the integral of P and dA would yield Force , right ?

Homework Equations

The Attempt at a Solution

You're confusing what is called the pressure prism by your text with a geometrical prism, a 3-dimensional body.

The volume of the pressure prism is equal to the magnitude of the force exerted by a hydrostatic pressure P applied over an area A, or to put it mathematically,

$$F = \int P\,dA$$

 

[werson tan]

You're confusing what is called the pressure prism by your text with a geometrical prism, a 3-dimensional body.

The volume of the pressure prism is equal to the magnitude of the force exerted by a hydrostatic pressure P applied over an area A, or to put it mathematically,

$$F = \int P\,dA$$

how can The volume of the pressure prism is equal to the magnitude of the force exerted by a hydrostatic pressure P applied over an area A ?

 

[SteamKing]

how can The volume of the pressure prism is equal to the magnitude of the force exerted by a hydrostatic pressure P applied over an area A ?

The magnitude of the force exerted over the area A by a hydrostatic pressure P is numerically equal to the volume of a prism which has a cross sectional area equal to A and a length equal to P units.

Consider a block which measures 1 m x 1 m x 1 m which is submerged in fresh water which has a density of 1000 kg / m³.

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If the top surface of the block is located 1 m below the surface of the water, the pressure P_TOP will be P_TOP = ρ g h = 1000 ⋅ 9.8 ⋅ 1 = 9,800 N/m².

P_TOP is constant over the entire area of the top of the block, which is A_TOP = 1 m². The volume of the pressure prism for the top of the block is V_{pressure prism} = P_TOP × A_TOP = 9,800 N/m² x 1 m² = 9,800 N. It just so happens that the force of the hydrostatic pressure acting on the top of the block is also 9,800 N.

A similar calculation can be made for the bottom surface of the block, but in this case h = 2 m and P_BOT = 19,600 N/m².

The sides of the block form a more complicated pressure prism, which has a trapezoidal cross section.

At the top of the prism, the hydrostatic pressure is 9,800 N/m², while the pressure on the bottom is 19,600 N/m², and the pressure at any depth in between the top and bottom of the block varies linearly. The average pressure on the sides, P_AVG = 14,700 N / m², can be used to calculate the force due to hydrostatic pressure acting on these surfaces.