Question

A barrel contains a 0.120-m layer of oil floating on water that is 0.250 m deep. The density of the oil is 600 kg/m\(^3\). (a) What is the gauge pressure at the oil\(-\)water interface? (b) What is the gauge pressure at the bottom of the barrel?

Short answer

(a) 705.6 Pa; (b) 3158.1 Pa.

Step-by-step solution

Understand the Problem

We have a barrel with 0.120 m of oil floating on top of 0.250 m of water. The density of the oil is given as 600 kg/m\(^3\). We need to find the gauge pressure at two points: (a) the oil-water interface, and (b) the bottom of the barrel.

Calculate Gauge Pressure at Oil-Water Interface

Gauge pressure is calculated using the equation for hydrostatic pressure: \( P = \rho gh \), where \( \rho \) is the density, \( g \) is the acceleration due to gravity (approximately 9.81 m/s\(^2\)), and \( h \) is the height of the fluid column. For the oil-water interface, we only consider the oil layer: \[ P_{\text{interface}} = (600 \text{ kg/m}^3)(9.81 \text{ m/s}^2)(0.120 \text{ m}) = 705.6 \text{ Pa} \].

Calculate Gauge Pressure at Bottom of the Barrel

For the bottom of the barrel, calculate the pressure due to both the oil and the water layers. First, find the pressure due to the water: \( P_{\text{water}} = \rho_{\text{water}}gh_{\text{water}} = (1000 \text{ kg/m}^3)(9.81 \text{ m/s}^2)(0.250 \text{ m}) = 2452.5 \text{ Pa} \). Add the pressure of the oil layer from the previous step: \[ P_{\text{bottom}} = 705.6 \text{ Pa} + 2452.5 \text{ Pa} = 3158.1 \text{ Pa} \].

Interpret the Results

The gauge pressure at the oil-water interface is 705.6 Pa, and at the bottom of the barrel, it is 3158.1 Pa. This means that the pressure increases as we go deeper into the fluid due to the additional weight of the liquid above.

Key concepts

Gauge Pressure

Gauge pressure is an essential concept when dealing with fluids, such as the oil and water in our barrel. Unlike absolute pressure, which includes atmospheric pressure, gauge pressure only considers the pressure relative to atmospheric pressure.
This is why when you measure gauge pressure, it can show as zero if the pressure is the same as the atmospheric pressure, and rise above zero when higher.

To calculate gauge pressure in any fluid at a specific point, we use the formula for hydrostatic pressure:

• \( P = \rho gh \)

Here, \( P \) represents the pressure we want to find; \( \rho \) is the density of the fluid; \( g \) is the gravitational acceleration (9.81 m/s\(^2\)); and \( h \) is the height of the fluid above that point.
When we calculated for the oil-water interface, we found \( 705.6 \) pascals.
For the bottom of the barrel, it was \( 3158.1 \) pascals, showing how pressure accumulates from the top down.

Density of Fluids

Density is a key factor in determining how pressure builds up in a fluid. It is defined as mass per unit volume, often given in kilograms per cubic meter (kg/m\(^3\)).
The density of a fluid influences the pressure it exerts at a specific depth because denser fluids have more mass and weight for the same volume.

In our example, we have two different fluids:

• The oil with a density of \( 600 \text{ kg/m}^3 \)
• The water with a typical density of \( 1000 \text{ kg/m}^3 \)

Because water is denser than the oil, when calculating the pressure at the barrel's bottom, we see that most of the pressure increase comes from the denser water layer.
In simple terms, denser fluids push or press down more due to their weight, creating more pressure.

Fluid Mechanics

Fluid mechanics is an important branch of physics that deals with the behavior and motion of liquids and gases.
Central to fluid mechanics are concepts such as pressure, density, and buoyancy, and understanding these helps us analyze how fluids behave in various scenarios.

In the case of the barrel with oil over water:

• We see fluid layers with distinct densities, showing stratification, common in fluid mechanics.
• Observing how the gauge pressure varies at different points in the fluid column is a basic exercise in fluid mechanics.

Mastery of fluid mechanics allows us to predict how different forces impact fluids, essential for fields ranging from engineering to meteorology.
It helps explain everyday phenomena and is applicable in designing systems like pipelines, dams, and even ships."