Problem 12 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

(a) 706.32 Pa; (b) 3158.82 Pa.

Step 1: Understand Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is calculated by considering only the pressure produced by the liquid at a given depth.

Step 2: Calculate Gauge Pressure at the Oil-Water Interface

The gauge pressure at the oil-water interface is due to the column of oil above it. Use the formula for pressure due to a liquid column: \( P = \rho_{\text{oil}} \cdot g \cdot h_{\text{oil}} \). Here, \( \rho_{\text{oil}} = 600 \, \text{kg/m}^3 \), \( g = 9.81 \, \text{m/s}^2 \), and \( h_{\text{oil}} = 0.120 \, \text{m} \).So, \( P = 600 \times 9.81 \times 0.120 = 706.32 \, \text{Pa} \).

Step 3: Calculate Gauge Pressure at the Bottom of the Barrel

The gauge pressure at the bottom is due to both the oil and the water columns. Calculate the pressure due to the water using the formula: \( P = \rho_{\text{water}} \cdot g \cdot h_{\text{water}} \). Here, \( \rho_{\text{water}} = 1000 \, \text{kg/m}^3 \) and \( h_{\text{water}} = 0.250 \, \text{m} \).So, the pressure due to water is \( 1000 \times 9.81 \times 0.250 = 2452.5 \, \text{Pa} \).Add the pressure from the oil layer, so the total gauge pressure at the bottom is \( 706.32 + 2452.5 = 3158.82 \, \text{Pa} \).

Key Concepts

Liquid Pressure

Liquid pressure refers to the pressure exerted by a liquid at a given point within itself or on the walls of a container. Imagine you are swimming underwater; the water above you presses down due to gravity. This pressure is a function of several things, including:

• The depth of the liquid: The deeper you go, the greater the pressure.
• The density of the liquid: Heavier liquids exert more pressure.
• Gravitational acceleration: Earth's gravity pulls the liquid downwards, creating pressure.

In our example with the barrel, the oil floating on the water creates pressure at the interface because of its weight and density. To calculate gauge pressure in any fluid, use the formula:\[ P = \rho \cdot g \cdot h \]where \( P \) is the pressure, \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, and \( h \) is the depth or height of the fluid column.

Density of Fluids

The density of fluids is a measure of how much mass is contained in a given volume. It's expressed as kilograms per cubic meter (kg/m³) in the metric system. Different liquids have different densities, which affect their behavior in fluid systems.

• Denser fluids exert more pressure because there is more mass in a given space.
• The density affects buoyancy, which is why oil floats on water—oil is less dense than water.

In the given exercise, the oil has a density of 600 kg/m³ while water has a density of 1000 kg/m³. Because of their densities, the calculations for gauge pressure vary between the oil and the water layers. Density is essential for calculating the pressure and understanding how fluids interact.

Fluid Statics

Fluid statics, also known as hydrostatics, is the study of fluids at rest. It is fundamental in understanding the behavior of liquid pressures in containers, such as barrels or bottles. Key principles include:

• Pascal's Law: Pressure exerted on a confined fluid is transmitted undiminished in all directions.
• Pressure increases with depth: As you move deeper in a fluid, the pressure increases because of the weight of the fluid above.
• These principles help us predict how liquids distribute within a structure like a barrel, where oil and water rest in layers.

In our exercise, by applying fluid statics, we recognize that the pressure at any point in the fluid depends on the properties of the fluid and the height of the fluid column above that point. Thus, to find the pressure at the bottom of the barrel, we account for both the oil and the underlying water, integrating the pressures exerted by each fluid layer.