Question

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.

Short answer

Plot \( R^2 \\ vs. \\ p_g \), find the line's slope \( m \) and intercept \( c \), then calculate \( h \) and \( \rho \) using \( m \) and \( c \).

Step-by-step solution

Understand the Relationship

We start by understanding the relationship between the horizontal distance the liquid travels, \( R \), and the gauge pressure, \( p_g \). When a fluid exits a hole at the bottom of a tank, the speed \( v \) can be described using Torricelli's Law: \( v = \sqrt{2gh} \), where \( h \) is the height of the liquid above the hole. Your task is to relate \( R \) to \( p_g \).

Relate Pressure, Velocity and Distance

The kinetic energy translated to the fluid can be represented by \( v^2 = \frac{2(p_g + \rho gh)}{\rho} \), where \( \rho \) is the fluid's density. When it exits, this velocity can be related to the horizontal distance \( R \) it travels. Since the hole is 0.5 m above the ground, use projectile motion: \( R = v \sqrt{\frac{2s}{g}} \), with \( s \) being the vertical drop (0.5 m).

Express \( R^2 \) in terms of Pressure and Height

From the projectile formula, \( R^2 = \left( \frac{2}{g} (p_g + \rho gh) \right) \). Rearrange to form \( y = mx + c \): \( R^2 = \frac{2}{\rho g}p_g \ + \frac{2h}{g} \). This shows that \( R^2 \) should be a straight line when plotted against \( p_g \).

Graph and Calculate Slope and Intercept

By graphing \( R^2 \) against \( p_g \), you should get a linear plot with a slope \( m = \frac{2}{\rho g} \) and y-intercept \( c = \frac{2h}{g} \). Use the graph data to find these values.

Solve for Height and Density

Using the slope \( m \) and intercept \( c \) from your graph, calculate \( \rho = \frac{2}{gm} \) and \( h = \frac{cg}{2} \), where \( g = 9.8 \text{ m/s}^2 \). Use your values to solve for \( \rho \) and \( h \).

Key concepts

Gauge Pressure

Gauge pressure is a key concept in fluid dynamics that is crucial for understanding many real-world applications, such as the behavior of liquids in a tank. It refers to the pressure relative to atmospheric pressure and can be positive or negative depending on whether it is above or below atmospheric levels. This is different from absolute pressure, which includes atmospheric pressure in its measurement.
Understanding gauge pressure helps to assess how much more (or less) pressure a liquid system has compared to the atmospheric surroundings.

• It's measured in Pascal (Pa).
• A positive gauge pressure indicates that the system has higher pressure than the atmosphere.
• When applied in the exercise scenario, the gauge pressure contributes to calculating the velocity of a liquid leaving a tank through a hole.

This is because the pressure difference is what drives the flow of the liquid, thus impacting its speed and the distance it will travel horizontally.

Torricelli's Law

Torricelli's Law is an important principle in fluid dynamics that describes the speed of a fluid flowing out of an opening under the influence of gravity. This law, formulated by Evangelista Torricelli, states that the speed (\(v\)) of efflux of a fluid under gravity through an orifice is equal to the speed that a body (dropped from the height (\(h\)) of the fluid surface above the opening) would acquire at the bottom. The formula is given by:\[v = \sqrt{2gh}\]where:

• \(g\) stands for the acceleration due to gravity (9.8 m/s²).
• \(h\) is the height of the fluid above the outlet.

Torricelli's Law is useful in the exercise for determining the velocity of the liquid exiting a hole in a tank. This velocity is critical in understanding how far the liquid will travel horizontally as a projectile.

Projectile Motion

Projectile motion is the motion of an object that is thrown or projected into the air and is subject only to the acceleration of gravity. In our scenario, the liquid exiting the tank through a hole follows a parabolic path, resembling projectile motion.
This type of motion is two-dimensional:

• Initial velocity: Horizontal component as derived from Torricelli’s Law.
• Vertical drop: Determined by gravity as the vertical component.

The projectile motion of the liquid tells us about the horizontal distance (\(R\)) it will travel:\[R = v \sqrt{\frac{2s}{g}}\]where:

• \(s\) is the vertical drop (0.5 m in this exercise).
• \(g\) is the acceleration due to gravity (9.8 m/s²).

This equation helps to correlate the horizontal distance with pressure, leading to the graphable relationship of (\(R^2\)) versus gauge pressure (\(p_g\)).

Density Calculation

Density calculation is pivotal for understanding how a fluid behaves in different conditions and is usually measured in kilograms per cubic meter (kg/m³). To find the density of the liquid in the tank, we analyze the slope from the graph of \(R^2\) versus \(p_g\).From the linear equation derived in the exercise, the density (\(\rho\)) of the fluid can be calculated using:\[\rho = \frac{2}{gm}\]where:

• \(m\) is the slope from the graph relating \(R^2\) and \(p_g\).
• \(g\) is the acceleration due to gravity (9.8 m/s²).

Determining the density is essential for characterizing the liquid's properties, which can help in identifying the liquid and ensuring safe handling, especially in industrial contexts like the investigation in the exercise.