Question

Two fluids, A and B, have densities of \(x\) and \(2 x\), respectively. They are tested independently to assess absolute pressure at varying depths. At what depths will the pressure below the surface of these two fluids be equal? A. Whenever the depth of fluid \(A\) is one-half that of fluid \(B\) B. Whenever the depth of fluid \(A\) equals that of fluid \(B\) C. Whenever the depth of fluid \(A\) is 2 times that of fluid \(B\) D. Whenever the depth of fluid \(A\) is 4 times that of fluid \(B\)

Short answer

C. Whenever the depth of fluid A is 2 times that of fluid B

Step-by-step solution

Understanding the Problem

We need to find the depth relationship between fluid A and fluid B such that the pressure below the surface of both fluids is equal.

Recall the Pressure Formula

The pressure at a depth in a fluid is given by the formula: \[ P = \rho g h \] Where: - \( P \) is the pressure - \( \rho \) is the density of the fluid - \( g \) is the acceleration due to gravity (constant) - \( h \) is the depth

Set Up Equations for Both Fluids

Using the pressure formula, set up the equations for both fluids: Fluid A: \[ P_A = x g h_A \] Fluid B: \[ P_B = 2 x g h_B \]

Set Equations Equal to Each Other

For the pressures to be equal, set \( P_A \) equal to \( P_B \): \[ x g h_A = 2 x g h_B \]

Simplify the Equation

Since the densities and gravity are constants, they can be canceled out: \[ h_A = 2 h_B \]

Interpret the Result

The result \( h_A = 2 h_B \) implies that the depth of fluid A must be twice the depth of fluid B for their pressures to be equal.

Key concepts

Fluid Mechanics

Fluid mechanics studies the behavior of fluids (liquids and gases) at rest and in motion. In this case, we focus on fluids at rest, also known as hydrostatics. When dealing with fluids at different depths, understanding how pressure changes with depth is crucial.

In hydrostatics, pressure increases with depth in a fluid due to the weight of the fluid above. This concept helps us analyze situations such as the one given: calculating depths where pressures of different fluids are equal. Understanding fluid mechanics provides clarity on diverse applications, from atmospheric science to hydraulics.

Absolute Pressure

Absolute pressure refers to the total pressure at a point within a fluid. It considers both the atmospheric pressure and the pressure due to the fluid’s depth. The formula for absolute pressure is:

// Inserted Latex formula below.

\[ P = P_{atm} + \rho g h \]

Here, \(P\) is the absolute pressure, \(P_{atm}\) the atmospheric pressure, \(\rho\) the fluid density, \(g\) the acceleration due to gravity, and \(h\) the depth.

For simplification when focusing on depth-related pressure, we often consider gauge pressure, which excludes atmospheric pressure.

In the exercise, we equate the pressures of two different fluids to find how their depths compare. Therefore, understanding absolute pressure and its components is essential.

Density Dependency

Density is a measure of a fluid's mass per unit volume, given by \(\rho\). The density of a fluid affects how pressure increases with depth. In our exercise, we have two fluids: Fluid A with density \(x\) and Fluid B with density \(2x\).

The more dense the fluid, the faster the pressure increases with depth. Therefore, Fluid B's pressure builds up quicker than Fluid A's. Consequently, to have equal pressure at certain depths, Fluid A must reach a greater depth than Fluid B.

Understanding density dependency can aid in various applications, including designing submerged structures and calculating buoyancy forces.

Pressure Formula

The fundamental formula for pressure in a fluid depends on depth, density, and gravity. The formula is:

\[ P = \rho g h \]

\(P\) is the pressure at depth, \(\rho\) is the fluid density, \(g\) is the gravitational acceleration, and \(h\) is the depth. In the given exercise, we use this formula to set up equations for two different fluids and find when their pressures are equal.

• Fluid A: \( P_A = x g h_A \)
• Fluid B: \( P_B = 2 x g h_B \)

Equating the pressures and simplifying:

\[ x g h_A = 2 x g h_B \]

leads us to:

\[ h_A = 2 h_B \]

This shows that Fluid A must be at twice the depth of Fluid B to have equal pressures. Mastery of the pressure formula aids in understanding various fluid-related problems.