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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

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

Step by step solution

01

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.
02

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: \[ P_{\text{interface}} = (600 \text{ kg/m}^3)(9.81 \text{ m/s}^2)(0.120 \text{ m}) = 705.6 \text{ Pa} \].
03

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: \[ P_{\text{bottom}} = 705.6 \text{ Pa} + 2452.5 \text{ Pa} = 3158.1 \text{ Pa} \].
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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."

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Most popular questions from this chapter

A solid aluminum ingot weighs 89 N in air. (a) What is its volume? (b) The ingot is suspended from a rope and totally immersed in water. What is the tension in the rope (the \(apparent\) weight of the ingot in water)?

Ballooning on Mars. It has been proposed that we could explore Mars using inflated balloons to hover just above the surface. The buoyancy of the atmosphere would keep the balloon aloft. The density of the Martian atmosphere is 0.0154 kg/m\(^3\) (although this varies with temperature). Suppose we construct these balloons of a thin but tough plastic having a density such that each square meter has a mass of 5.00 g. We inflate them with a very light gas whose mass we can ignore. (a) What should be the radius and mass of these balloons so they just hover above the surface of Mars? (b) If we released one of the balloons from part (a) on earth, where the atmospheric density is 1.20 kg/m\(^3\), what would be its initial acceleration assuming it was the same size as on Mars? Would it go up or down? (c) If on Mars these balloons have five times the radius found in part (a), how heavy an instrument package could they carry?

In seawater, a life preserver with a volume of 0.0400 m\(^3\) will support a 75.0-kg person (average density 980 kg/m\(^3\)), with 20% of the person's volume above the water surface when the life preserver is fully submerged. What is the density of the material composing the life preserver?

A liquid flowing from a vertical pipe has a definite shape as it flows from the pipe. To get the equation for this shape, assume that the liquid is in free fall once it leaves the pipe. Just as it leaves the pipe, the liquid has speed \(\upsilon$$_0\) and the radius of the stream of liquid is \({r_0}\). (a) Find an equation for the speed of the liquid as a function of the distance \(y\) it has fallen. Combining this with the equation of continuity, find an expression for the radius of the stream as a function of \(y\). (b) If water flows out of a vertical pipe at a speed of 1.20 m/s, how far below the outlet will the radius be one-half the original radius of the stream?

The Environmental Protection Agency is investigating an abandoned chemical plant. A large, closed cylindrical tank contains an unknown liquid. You must determine the liquid's density and the height of the liquid in the tank (the vertical distance from the surface of the liquid to the bottom of the tank). To maintain various values of the gauge pressure in the air that is above the liquid in the tank, you can use compressed air. You make a small hole at the bottom of the side of the tank, which is on a concrete platform\(-\)so the hole is 50.0 cm above the ground. The table gives your measurements of the horizontal distance \(R\) that the initially horizontal stream of liquid pouring out of the tank travels before it strikes the ground and the gauge pressure \({p_g}\) of the air in the tank. (a) Graph \({R^2}\) as a function of \({p_g}\). Explain why the data points fall close to a straight line. Find the slope and intercept of that line. (b) Use the slope and intercept found in part (a) to calculate the height \(h\) (in meters) of the liquid in the tank and the density of the liquid (in kg/m\(^3\)). Use \(g\) \(=\) 9.80 m/s\(^2\). Assume that the liquid is nonviscous and that the hole is small enough compared to the tank's diameter so that the change in h during the measurements is very small.

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