Problem 29 - Basic Technical Mathematics with Calculus

Question

A watertight cubical box with an edge of \(2.00 \mathrm{m}\) is suspended in water such that the top surface is \(1.00 \mathrm{m}\) below water level. Find the total force on the top of the box and the total force on the bottom of the box. What meaning can you give to the difference of these two forces?

Step-by-Step Solution

Answer

The force on the top is 39240 N, on the bottom is 117720 N; the difference is the buoyant force (78480 N).

Step 1: Determine the Position of Both Surfaces

The top surface of the cube is at a depth of \(1.00 \; \mathrm{m}\) below the water level. Since the cube is \(2.00 \; \mathrm{m}\) in size, the bottom surface is \(1.00 + 2.00 = 3.00 \; \mathrm{m}\) below the water surface.

Step 2: Calculate Pressure on the Top Surface

The pressure on the top surface, which is \(1.00 \; \mathrm{m}\) below the water, is given by \(P = \rho \cdot g \cdot h\), where \(\rho\) is the water density \(1000 \; \mathrm{kg/m^3}\), \(g\) is the acceleration due to gravity \(9.81 \; \mathrm{m/s^2}\), and \(h\) is the depth. Thus, \(P_{\text{top}} = 1000 \cdot 9.81 \cdot 1.00 = 9810 \; \mathrm{N/m^2}\).

Step 3: Calculate Force on the Top Surface

Force is given by \(F = P \cdot A\), where \(A\) is the area of the surface. The area of the top of the cube is \(2.00 \; \mathrm{m} \times 2.00 \; \mathrm{m} = 4.00 \; \mathrm{m^2}\). The force on the top is \(F_{\text{top}} = 9810 \cdot 4.00 = 39240 \; \mathrm{N}\).

Step 4: Calculate Pressure on the Bottom Surface

The pressure on the bottom surface, \(3.00 \; \mathrm{m}\) below the water, is \(P_{\text{bottom}} = 1000 \cdot 9.81 \cdot 3.00 = 29430 \; \mathrm{N/m^2}\).

Step 5: Calculate Force on the Bottom Surface

Using the same area, the force on the bottom is \(F_{\text{bottom}} = 29430 \cdot 4.00 = 117720 \; \mathrm{N}\).

Step 6: Calculate the Difference in Forces

The difference in force on the top and bottom surfaces is due to the buoyant force acting on the box. The difference is \(F_{\text{bottom}} - F_{\text{top}} = 117720 - 39240 = 78480 \; \mathrm{N}\). This force is equal to the weight of the water displaced by the box, hence it reflects the buoyant force.

Key Concepts

Buoyant Force

When an object is submerged in a fluid, it experiences an upward force. This force is known as the buoyant force. The buoyant force acts opposite to the direction of gravity, helping things float or appear lighter in water.
The buoyant force results from the pressure difference between the top and bottom surfaces of the submerged object. The deeper the object, the greater the pressure difference. This is why the bottom of a submerged object experiences more pressure than the top.
The buoyant force is equal to the weight of the fluid that the object displaces. For a cubic box submerged in water, like in our exercise, the buoyant force can be calculated by finding the difference between the forces on the top and bottom surfaces. In the provided exercise, this difference is 78480 N, which precisely equals the weight of the water displaced by the box.

• Helps explain why objects float or sink.
• Depends on fluid density and gravitational pull.
• Essential for understanding fluid dynamics.

Pressure Calculation

Pressure is a key concept in fluid mechanics that helps us understand how forces are distributed over surfaces. In fluids, pressure is the force per unit area exerted by the fluid. When calculating pressure in a fluid, we rely on the equation: \[ P = \rho \cdot g \cdot h \]where:

• \( \rho \) is the fluid density.
• \( g \) is the acceleration due to gravity.
• \( h \) is the depth below the fluid surface.

For the cube in the exercise, we used this formula to determine the pressure on the cube's surfaces. For the top at 1 meter depth, and the bottom at 3 meters depth, the pressure was calculated as 9810 N/m² and 29430 N/m² respectively. The difference in these pressures highlights how depth affects the force experienced by submerged surfaces.

Force on Submerged Surfaces

When an object is submerged in a fluid, it experiences forces on its surfaces due to the fluid's pressure. These forces can be calculated using the formula:
\[ F = P \cdot A \]
where

• \( F \) is the force.
• \( P \) is the pressure.
• \( A \) is the area of the surface.

The area of the surfaces in our exercise is obtained from the square side of the cube, yielding 4 m² for both top and bottom. By multiplying the area by the respective pressures (9810 N/m² for the top and 29430 N/m² for the bottom), we calculated forces of 39240 N on the top and 117720 N on the bottom.
This demonstrates how pressure translates into force on submerged surfaces. Notably, these forces are not balanced, leading to the buoyant effect. The differing forces demonstrate the essential principles of hydrostatics and are crucial for solving various engineering and physics problems.