Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the total force exerted by the water on all sides of a cube of edge length 2 feet if its top is horizontal and 100 feet below the surface of a lake.

Step-by-Step Solution

Answer

The total force exerted by the water on the cube is approximately 806,400 lb.

Step 1: Understand the Pressure Formula

The pressure exerted by a fluid at a given depth is calculated using the formula \( P = \rho g h \), where \( P \) is the pressure, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity, and \( h \) is the depth of the fluid.

Step 2: Calculate the Pressure on Top of the Cube

Since the top of the cube is 100 feet below the surface, the pressure on it is \( P_{top} = \rho g \times 100 \). For water, \( \rho \approx 62.4 \text{ lb/ft}^3 \) and \( g \approx 32.2 \text{ ft/s}^2 \), so \( P_{top} = 62.4 \times 32.2 \times 100 \).

Step 3: Calculate the Pressure on the Sides of the Cube

The sides of the cube have pressure varying linearly from 100 feet to 102 feet below the surface. The average depth is 101 feet, so the pressure on the sides is \( P_{side} = \rho g \times 101 \). Hence, \( P_{side} = 62.4 \times 32.2 \times 101 \).

Step 4: Calculate the Area of Each Face of the Cube

Each face of the cube has an area of \( A = 2^2 = 4 \text{ ft}^2 \).

Step 5: Calculate the Force on the Top and Bottom Faces

Force is calculated using the formula \( F = P \times A \). For the top face, \( F_{top} = P_{top} \times 4 \). The bottom face is 2 feet lower, at 102 feet below the surface, so \( P_{bottom} = \rho g \times 102 \), and \( F_{bottom} = P_{bottom} \times 4 \).

Step 6: Calculate the Force on the Side Faces

The force on the sides is uniform over each side, calculated as \( F_{side} = P_{side} \times 4 \). There are four sides, so the total force on the sides is \( 4 \times F_{side} \).

Step 7: Sum All the Forces

Total force, \( F_{total} = F_{top} + F_{bottom} + 4 \times F_{side} \). Substitute the calculated forces from Steps 5 and 6 and add them together to find the total force.

Step 8: Calculate the Total Force

Substituting the calculated values: \( F_{total} = (62.4 \times 32.2 \times 100 \times 4) + (62.4 \times 32.2 \times 102 \times 4) + 4 \times (62.4 \times 32.2 \times 101 \times 4) \). Solving these gives the total force exerted by the water on the cube.

Key Concepts

Fluid Pressure

Fluid pressure is the force exerted by a fluid per unit area. You encounter fluid pressure when a fluid, like water, is exerting a force on an object. This concept is vital in hydrostatics, where we analyze fluids at rest.
Fluid pressure in a liquid depends on several factors:

• Density of the Fluid: Heavier fluids exert more pressure.
• Depth: The deeper an object is submerged, the greater the pressure.
• Gravity: The force that pulls the fluid towards Earth, creating pressure.

This relationship is perfectly described with the equation: \( P = \rho g h \), where \( P \) symbolizes the pressure, \( \rho \) represents the fluid's density, \( g \) signifies gravitational acceleration, and \( h \) denotes the depth at which the pressure is calculated. This equation allows us to calculate the pressure at any point under the water, showing why the pressure increases deeper under the surface.

Force Calculation

Force calculation in the context of hydrostatics involves figuring out how much total push or pull the fluid exerts on an object.
We can determine the force using the formula: \( F = P \times A \), where \( F \) stands for force, \( P \) is the pressure, and \( A \) the area. Understanding this helps you find out how much force an object - say a submerged cube - experiences from all sides.
For example, if a cube's face measures 4 square feet and experiences a pressure, we multiply the face area by the pressure to find the force on that face. Repeating this for each face allows us to compute the total force on the entire object. The simple multiplication of pressure and area thus provides insight into how large the impact of the water on the object will be.

Pressure Formula

The pressure formula, \( P = \rho g h \), serves as a cornerstone in determining how much pressure a fluid exerts at any given depth. This equation is not just essential for understanding pressure-related queries but also crucial in solving numerous real-world problems related to underwater engineering or diving.
Let's break down each part:

• \( \rho \) is the density, and it tells us how much mass is in a given volume of the fluid.
• \( g \) is the acceleration due to gravity, generally about 32.2 ft/s² on Earth.
• \( h \) indicates how deep the object is in the fluid.

The equation is relatively simple, yet it effectively captures the idea that pressure increases with depth and the fluid's density. This basic formula helps engineers and scientists predict and manage situations involving fluids.

Underwater Mechanics

Underwater mechanics examines how forces interact with objects submerged in water, providing crucial insights into designing undersea structures and equipment. Understanding these mechanics can help you appreciate why things behave differently underwater compared to on land.
In underwater mechanics, several elements are essential:

• Buoyancy: The upward force water exerts that makes submerged objects lighter.
• Pressure Gradients: How pressure changes with depth, impacting the forces on submerged surfaces.
• Material Strength: Materials must withstand underwater conditions, like pressure differences.

The concept of underwater mechanics allows us to calculate the stresses objects endure underwater and design systems that remain effective and secure in these challenging environments. Each factor ensures stability and safety, crucial for applications like underwater diving, submarines, and underwater tunnels.