Problem 29 - Physics

Question

A 1.00-m-tall container is filled to the brim, partway with mercury and the rest of the way with water. The container is open to the atmosphere. What must be the depth of the mercury so that the absolute pressure on the bottom of the container is twice the atmospheric pressure?

Step-by-Step Solution

Answer

The mercury depth must be approximately 0.76 m.

Step 1: Understand the Problem

We have a container filled with mercury and water to a total height of 1 meter. The problem asks us to find the depth of mercury such that the absolute pressure at the bottom is twice the atmospheric pressure.

Step 2: Identify Key Formulas

To find the pressure, use the formula for liquid pressure: \( P = \rho gh \), where \( \rho \) is the density, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid.

Step 3: Express Total Absolute Pressure

The absolute pressure at the bottom of the container is the sum of the atmospheric pressure, the pressure due to the mercury, and the pressure due to the water. Thus, \( P_\text{bottom} = P_\text{atm} + \rho_\text{Hg}g h_\text{Hg} + \rho_\text{water}g (1 - h_\text{Hg}) \).

Step 4: Set Up the Equation for Twice the Atmospheric Pressure

We need \( P_\text{bottom} = 2P_\text{atm} \). Substitute the expression for \( P_\text{bottom} \): \[ 2P_\text{atm} = P_\text{atm} + \rho_\text{Hg}g h_\text{Hg} + \rho_\text{water}g (1 - h_\text{Hg}) \].

Step 5: Simplify the Equation

Rearrange the equation to focus on the term involving \( h_\text{Hg} \): \[ P_\text{atm} = \rho_\text{Hg}g h_\text{Hg} + \rho_\text{water}g (1 - h_\text{Hg}) \].

Step 6: Isolate the Mercury Depth

Solve for \( h_\text{Hg} \): \[ h_\text{Hg} = \frac{P_\text{atm} - \rho_\text{water}g}{(\rho_\text{Hg} - \rho_\text{water})g} \].

Step 7: Substitute Known Values

Use \( \rho_\text{Hg} = 13,600 \text{ kg/m}^3 \), \( \rho_\text{water} = 1,000 \text{ kg/m}^3 \), \( g = 9.81 \text{ m/s}^2 \), and \( P_\text{atm} = 101,325 \text{ Pa} \): \[ h_\text{Hg} = \frac{101,325 - 1,000 \cdot 9.81}{(13,600 - 1,000) \cdot 9.81} \].

Step 8: Calculate the Mercury Depth

Plug in the values and calculate: \( h_\text{Hg} \approx 0.76 \text{ m} \).

Key Concepts

Density of Mercury

The density of mercury is significantly higher than that of many common liquids, like water. Specifically, mercury has a density of \(13,600 \ \text{kg/m}^3\). This means it is very heavy per unit volume. When discussing density:

• Density is defined as mass per unit volume.
• High density indicates that there is a lot of mass in a small volume.

Since mercury is much denser than water, it generates a higher pressure at any given depth. In the context of a liquid-filled container, this means mercury contributes significantly to the total pressure at the bottom. The heavy nature of mercury is utilized in various applications, especially in measuring devices like barometers.

Density of Water

Water is a standard reference when discussing fluid densities, with a density of \(1,000 \ \text{kg/m}^3\). This is considered relatively low, especially compared to substances like mercury. Here's why density matters:

• Water’s lower density means it exerts less pressure at a given depth compared to a denser fluid.
• This property of water makes it less effective in pressurizing the bottom of a container compared to denser substances.

In our exercise, the remaining space in the container that isn't filled with mercury is filled with water. This means it assists in adding to the overall pressure, though its contribution is smaller compared to mercury due to its lower density.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It's an essential concept for understanding scenarios involving fluids at rest:

• The formula for hydrostatic pressure is \(P = \rho gh\), with \( \rho \) as density, \( g \) as gravitational acceleration, and \( h \) as height or depth.
• This pressure increases with depth as more fluid weight pushes down from above.

With mercury and water in the container, each contributes to the total hydrostatic pressure at the container's bottom. Mercury, being denser, contributes more significantly to this pressure. The total hydrostatic pressure is the sum contribution from both fluids.

Atmospheric Pressure

Atmospheric pressure is the pressure exerted by the weight of the atmosphere above us. It is approximately \(101,325 \ \text{Pa}\) at sea level. When solving fluid problems such as the one given, atmospheric pressure forms part of the absolute pressure at the bottom of the container.

• Absolute pressure is the sum of atmospheric pressure and any additional pressure exerted by the fluid(s) in the container.
• In the problem where the aim is to make the absolute pressure twice the atmospheric pressure, atmospheric pressure provides the baseline value that must be exceeded by the pressure exerted by the contained fluids.

This concept is critical in ensuring that the calculated pressures unlike the surrounding air are accurately considered in real-world applications. Atmospheric pressure will always play a role unless a container is in a vacuum.