Introduction

In physics, the concept of floating and sinking is essential when studying the behavior of objects in fluids, particularly liquids like water. Understanding why some objects float while others sink is crucial for various applications, including shipbuilding, buoyancy, and even everyday activities like swimming.

When an object is placed in a fluid, it experiences an upward force called buoyancy, which opposes the force of gravity pulling it downward. Whether an object floats or sinks depends on the relationship between its weight and the buoyant force acting on it.

Archimedes' Principle

Archimedes' Principle states that the buoyant force acting on an object is equal to the weight of the fluid displaced by the object. Mathematically, this can be represented as:

$$ F_{\text{buoyant}} = \rho_{\text{fluid}} \cdot V_{\text{displaced}} \cdot g $$

Where:

• $ F_{\text{buoyant}} $ is the buoyant force,
• $ \rho_{\text{fluid}} $ is the density of the fluid,
• $ V_{\text{displaced}} $ is the volume of the fluid displaced by the object,
• $ g $ is the acceleration due to gravity.

Example 1

A wooden block of density $ 800 , \text{kg/m}^3 $ and volume $ 0.02 , \text{m}^3 $ is submerged in water. Calculate the buoyant force acting on the block.

Given:

• $ \rho_{\text{water}} = 1000 , \text{kg/m}^3 $
• $ g = 9.81 , \text{m/s}^2 $

Solution: The volume of water displaced is the same as the volume of the block: $$ V_{\text{displaced}} = 0.02 , \text{m}^3 $$

Calculate the buoyant force: $$ F_{\text{buoyant}} = \rho_{\text{water}} \cdot V_{\text{displaced}} \cdot g $$ $$ F_{\text{buoyant}} = 1000 \times 0.02 \times 9.81 = 196.2 , \text{N} $$

Therefore, the buoyant force acting on the wooden block is $ 196.2 , \text{N} $.

Density and Relative Density

• Density: Density is the mass per unit volume of a substance and is represented by the symbol $ \rho $. The formula for density is: $$ \rho = \frac{m}{V} $$

• Relative Density: Relative density, also known as specific gravity, compares the density of a substance to the density of water. It has no units and is a pure number. The relative density of a substance is given by: $$ \text{Relative Density} = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}} $$

Example 2

Calculate the relative density of a substance with a density of $ 1200 , \text{kg/m}^3 $.

Given:

• $ \rho_{\text{substance}} = 1200 , \text{kg/m}^3 $
• $ \rho_{\text{water}} = 1000 , \text{kg/m}^3 $

Solution: Using the formula for relative density: $$ \text{Relative Density} = \frac{1200}{1000} = 1.2 $$

Therefore, the relative density of the substance is 1.2.

Upthrust and Weight

• Upthrust: Upthrust is the force exerted by a fluid on an object immersed in it and is equal to the weight of the fluid displaced by the object. It is synonymous with the buoyant force.

• Weight: The weight of an object is the force acting on it due to gravity. It can be calculated using the formula: $$ W = m \cdot g $$

Example 3

A metal block has a mass of $ 5 , \text{kg} $. Calculate its weight and the upthrust when submerged in water.

Given:

• $ m = 5 , \text{kg} $
• $ g = 9.81 , \text{m/s}^2 $
• $ \rho_{\text{water}} = 1000 , \text{kg/m}^3 $

Solution: Calculate the weight of the metal block: $$ W = m \cdot g = 5 \times 9.81 = 49.05 , \text{N} $$

Since the weight of the water displaced is equal to the weight of the block, the upthrust is also $ 49.05 , \text{N} $.

Stability and Equilibrium

• Stability: An object is stable if it returns to its original position when disturbed. In the context of floating and sinking, stability is crucial to prevent capsizing or tipping over.

• Equilibrium: An object is in equilibrium when the sum of all forces acting on it is zero. This means that the object is not accelerating in any direction.

Example 4

Explain why a ship is designed with a wide base.

Explanation: A ship is designed with a wide base to increase its stability. A wider base lowers the center of gravity, making it harder for the ship to tip over. This design reduces the chances of capsizing, especially in rough waters.

Common Mistakes

• Neglecting the buoyant force: Some students forget to consider the buoyant force acting on an object when determining whether it will float or sink. Always remember that the buoyant force opposes the weight of the object.

• Confusing weight and upthrust: Weight is the force due to gravity acting on an object, while upthrust is the force exerted by a fluid on the object. Be clear about the distinction between these two forces.

• Misinterpreting stability: Understand that stability involves the ability of an object to return to its original position after being disturbed. A wider base contributes to greater stability.

Key Points

• Archimedes' Principle relates the buoyant force to the weight of the fluid displaced.
• Density is the mass per unit volume of a substance, while relative density compares the density of a substance to that of water.
• Upthrust is the force exerted by a fluid on an object immersed in it, equal to the weight of the fluid displaced.
• Stability is crucial for preventing tipping over, and equilibrium occurs when the sum of all forces on an object is zero.

Practice Questions

1. A wooden block of volume $ 0.05 , \text{m}^3 $ and density $ 600 , \text{kg/m}^3 $ is placed in a container of water. Calculate the buoyant force acting on the block.

Answer: The buoyant force is $ 294.3 , \text{N} $.

2. Determine the relative density of a substance with a density of $ 800 , \text{kg/m}^3 $.

Answer: The relative density is 0.8.

3. If an object has a weight of $ 25 , \text{N} $ when in air, what will be its weight when submerged in water?

Answer: The weight when submerged will be $ 15 , \text{N} $.

4. Explain why a boat with a low center of gravity is more stable than one with a high center of gravity.

Answer: A boat with a low center of gravity is more stable because it is less likely to tip over due to the lower point where the weight of the boat is concentrated.

5. A metal block with a mass of $ 8 , \text{kg} $ is submerged in a liquid. If the upthrust acting on the block is $ 60 , \text{N} $, calculate the density of the liquid.

Answer: The density of the liquid is $ 750 , \text{kg/m}^3 $.