Problem 23 - College Physics

Question

A barrel contains a 0.120 \(\mathrm{m}\) layer of oil of density 600 \(\mathrm{kg} / \mathrm{m}^{3}\) floating on water that is 0.250 \(\mathrm{m}\) deep. (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: Understanding Gauge Pressure

Gauge pressure is the pressure in a system above atmospheric pressure. It is calculated using the formula: \( P_g = \rho \cdot g \cdot h \), where \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( h \) is the height of the fluid column.

Step 2: Calculate Pressure at Oil-Water Interface

First, calculate the gauge pressure at the oil-water interface, caused by the oil layer. The density of the oil is given as 600 kg/m³ and its thickness as 0.120 m. Using the formula for gauge pressure:\[ P_o = 600 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.120 \, \text{m} \]\[ P_o = 706.32 \, \text{Pa} \]So, the gauge pressure at the oil-water interface is 706.32 Pa.

Step 3: Calculate Pressure at the Bottom of the Barrel

For the bottom of the barrel, we need to consider both the oil and the water pressures. First, calculate the pressure due to the water layer:The density of water is 1000 kg/m³, and the water depth is 0.250 m:\[ P_w = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.250 \, \text{m} \]\[ P_w = 2452.5 \, \text{Pa} \]Now, add the pressures from the oil and water to find the total gauge pressure:\[ P_{total} = P_o + P_w = 706.32 \, \text{Pa} + 2452.5 \, \text{Pa} = 3158.82 \, \text{Pa} \]Thus, the gauge pressure at the bottom of the barrel is 3158.82 Pa.

Key Concepts

Gauge Pressure

Gauge pressure is a concept in fluid mechanics that helps us understand the pressure within a system relative to the surrounding atmospheric pressure. This type of pressure, unlike absolute pressure, does not include atmospheric pressure in its measurement. It can be expressed with the formula \[ P_g = \rho \cdot g \cdot h \] where:

• \( P_g \) is the gauge pressure.
• \( \rho \) represents the density of the fluid.
• \( g \) is the acceleration due to gravity.
• \( h \) is the height of the fluid column.

Using gauge pressure, engineers and scientists can determine the pressure exerted by the fluid when put in context with the surrounding atmosphere. It's an invaluable tool especially in fields like hydraulics and pneumatics.
Understanding the concept of gauge pressure is vital when analyzing scenarios involving different layers of fluids, as it allows for the calculation of pressures at various depths.

Density Calculation

Calculating density is crucial in fluid mechanics for determining how different fluids interact in a container. Density, symbolized by \( \rho \), is calculated by the formula \[ \rho = \frac{m}{V} \]where:

• \( \rho \) represents the density.
• \( m \) stands for mass.
• \( V \) represents volume.

This calculation allows for understanding the relative heaviness of a fluid per unit of volume and plays a key role in scenarios involving layered fluids like those in the original exercise.
Knowing the density tells us how a fluid will behave under pressure and how it will interact when layered with another fluid. When fluids are layered, the fluid with greater density will be found below a less dense fluid. This principle explains why oil floats on water, as demonstrated in the exercise.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at a given depth or height. It is an essential concept when analyzing fluids in static situations. The formula for calculating hydrostatic pressure is quite similar to that of gauge pressure:\[ P = \rho \cdot g \cdot h \]where:

• \( P \) is hydrostatic pressure.
• \( \rho \) is the density of the fluid.
• \( g \) is gravitational acceleration.
• \( h \) is the depth in the fluid.

Hydrostatic pressure increases with depth, which is why the pressure at the bottom of the barrel in the original problem is greater than at the oil-water interface.
Understanding hydrostatic pressure helps us predict how fluids will behave in tanks and containers, and it is an integral part of planning for construction, engineering, and environmental studies where fluid management is necessary.

Layered Fluids in Physics

Layered fluids occur when two or more fluids with different densities are placed on top of one another without mixing. When dealing with these situations in physics, it's important to understand how each layer contributes to the total pressure within the system.
In the exercise above, two fluids—oil and water—are layered within a barrel. This scenario requires calculating separate gauge pressures for each layer and then summing them to determine the overall pressure at the bottom of the barrel.
Key points to understand about layered fluids include:

• The fluid with the lower density will float, like the oil on water, because it weighs less per unit volume.
• Each fluid layer contributes to the pressure felt below it, more so with greater height and density.
• The total pressure at a certain depth is the sum of pressures due to all the layers above.

Understanding these concepts is crucial when considering practical applications involving multiple fluids, such as oil containment or aquifer management.