Problem 3 - Applied Physics

Question

A rock weighs \(25.7 \mathrm{~N}\) in air and \(21.8 \mathrm{~N}\) in water. What is the buoyant force of the water?

Step-by-Step Solution

Answer

The buoyant force is 3.9 N.

Step 1: Identify Forces in Air and Water

Start by identifying the force of the rock when weighed in air and when submerged in water. In air, the rock's weight is given as 25.7 N, and in water, its weight is 21.8 N. These forces are due to gravity and buoyancy.

Step 2: Define Buoyant Force Concept

Recall that the buoyant force is the upward force the fluid (water, in this case) exerts on an object submerged in it. This is the reason the rock appears lighter in water compared to air.

Step 3: Apply the Buoyant Force Formula

The buoyant force can be calculated as the difference between the object's weight in air and its weight in water. Using the formula: \(F_b = W_{air} - W_{water}\), substitute the values \(W_{air} = 25.7 \, \text{N}\) and \(W_{water} = 21.8 \, \text{N}\).

Step 4: Calculate the Buoyant Force

Subtract the weight of the rock in water from its weight in air: \(F_b = 25.7 \, \text{N} - 21.8 \, \text{N} = 3.9 \, \text{N}\). This is the buoyant force exerted by the water.

Key Concepts

Archimedes' Principle

Archimedes' principle is a key idea in understanding why objects float or sink in fluids. This principle states that any object wholly or partially submerged in a fluid experiences an upward force. This force, called buoyancy, is equal to the weight of the fluid that the object displaces.
In the context of our exercise, when the rock is submerged in water, it displaces a certain amount of water, which results in a buoyant force. The difference in weight measurements — 25.7 N in air versus 21.8 N in water — is due to this buoyant force exerted by the displaced water.
When applying Archimedes' principle:

• The rock appears lighter in water because the buoyant force counteracts some of the gravitational pull.
• The amount of water displaced directly determines the magnitude of this upward force.

This principle helps us calculate the buoyant force just as demonstrated in the solution, by finding the weight difference of the rock in air and in water.

Physics Problems

Solving physics problems involves both conceptual understanding and methodical calculations. It's important to understand the underlying physical principles, as seen in our exercise involving weights and buoyancy.
The problem we solved is a classic example of a straightforward physics problem where we use observed data to find unknown quantities, such as buoyant force. Here's a glimpse into problem-solving strategies:

• Begin by identifying given data clearly, like the weights provided in the problem.
• Recall relevant physics principles — in this case, Archimedes' principle.
• Use a formula or equation that relates the knowns to the unknowns. Here, we used the difference in weights.
• Finally, carry out the calculations with attention to units and precision.

Working through these steps helps clarify the solution and enhances understanding of the physical concepts at play.

Fluid Mechanics

Fluid mechanics is the branch of physics that studies the behavior of fluids (liquids and gases) and how they interact with various forces. This broad field covers many topics, but buoyancy is one of its essential concepts. Buoyancy is a crucial aspect of our understanding of fluid dynamics, particularly in the context of submerged objects.
In fluid mechanics, the density of the fluid often influences the magnitude of the buoyant force. Although not directly mentioned in our problem, this concept is critical in more complex scenarios:

• The more dense a fluid, the greater the buoyant force it can exert on an object.
• Understanding this helps in designing ships and submarines, where buoyancy is vital for flotation.
• Buoyancy principles are utilized in various engineering and environmental applications.

Through the lens of fluid mechanics, our simple rock-and-water problem illustrates this fundamental interaction between objects and the fluids they are submerged in.