Problem 15 - Applied Physics

Question

What is the volume (in \(\mathrm{m}^{3}\) ) of the water displaced by a submerged air tank that is acted on by a buoyant force of \(7.50 \times 10^{4} \mathrm{~N}\) ?

Step-by-Step Solution

Answer

The volume of water displaced is approximately 7.64 m³.

Step 1: Understanding Buoyant Force

The buoyant force is equal to the weight of the fluid displaced by the submerged object. This relation is described by Archimedes' principle. Hence, \( F_b = \rho \cdot V \cdot g \), where \( F_b \) is the buoyant force, \( \rho \) is the density of the fluid, \( V \) is the volume of the fluid displaced, and \( g \) is the acceleration due to gravity.

Step 2: Identify Physical Constants

For water, the density \( \rho \) is approximately \( 1000 \ \mathrm{kg/m^3} \) and the acceleration due to gravity \( g \) is \( 9.81 \ \mathrm{m/s^2} \). These values will be used in the equation to find the volume of water displaced.

Step 3: Rearrange the Formula for Volume

Rearrange the formula to solve for \( V \): \( V = \frac{F_b}{\rho \cdot g} \). This gives us the volume of the water displaced in cubic meters.

Step 4: Substitute Known Values

Substitute the known values into the rearranged formula: \( F_b = 7.50 \times 10^4 \ \mathrm{N} \), \( \rho = 1000 \ \mathrm{kg/m^3} \), and \( g = 9.81 \ \mathrm{m/s^2} \). Insert these into the formula: \( V = \frac{7.50 \times 10^4}{1000 \times 9.81} \).

Step 5: Calculate the Volume

Carry out the division to find the volume. \( V = \frac{7.50 \times 10^4}{9810} \). Calculating this gives approximately \( 7.64 \ \mathrm{m^3} \).

Key Concepts

Archimedes' Principle

When you submerge an object in a fluid, such as water, it experiences an upward force. This force is known as the buoyant force and is equal to the weight of the fluid that the object displaces. This concept was discovered by the ancient Greek mathematician Archimedes. Archimedes' Principle is crucial in understanding why objects float or sink. It states that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. Thus, the greater the volume of the water displaced, the greater the buoyant force acting upon the object.

By applying Archimedes' Principle, we can solve many real-life problems, such as determining whether a ship will float and calculating the volume of submerged objects. In practical situations, it is particularly useful in everyday applications, ranging from designing ships to measuring the density of an irregularly shaped object using water displacement.

Density of Water

Density is a measure of how much mass is contained in a given volume. For water, specifically fresh water, this density is typically about 1000 kilograms per cubic meter ( \(1000 \, \mathrm{kg/m^3}\) ). Understanding the density of the fluid is essential because it directly affects the calculation of buoyant force.

In problems involving buoyancy, the density of the fluid is a key parameter. In our example, since the object is submerged in water, knowing the water's density allows us to accurately calculate how much water is displaced and therefore the buoyant force acting on the submerged object. This is an essential concept in physics problem-solving scenarios involving fluids.

Volume of Displacement

The volume of displacement refers to the amount of fluid that is moved aside by a submerged object. This volume is critical in calculating the buoyant force. According to Archimedes' Principle, the buoyant force matches the weight of the fluid displaced. Therefore, the greater the volume of water displaced, the greater the buoyant force.

To find this volume, you can rearrange the formula for buoyant force: \[ V = \frac{F_b}{\rho \cdot g} \] , where \(F_b\) is the buoyant force, \(\rho\) is the density of the fluid, and \(g\) is the acceleration due to gravity. By understanding the volume of displacement, you can predict whether an object will float or sink, or even compute the precise volume of fluid displaced.

Physics Problem Solving

When faced with physics problems, especially those involving buoyancy, a step-by-step approach is useful. Start by thoroughly understanding the problem requirements. Identify what is known and what needs to be determined.

Follow these steps to solve buoyancy-related problems:

• Understand the principles involved, like Archimedes’ Principle.
• Identify all known values from the problem statement (e.g., buoyant force, density, gravity).
• Rearrange key formulas to solve for the unknown variable, such as the volume of displaced fluid.
• Substitute the known values into the equation and compute the result.
• Evaluate your result to ensure it makes practical sense.

This systematic approach not only helps in solving the current problem but also builds a solid foundation for tackling more complex physics problems in the future.