1.2 Density

1.2 Density

Density: A Fundamental Concept in Fluid Mechanics

When dealing with fluids, we require a property that measures the relationship between mass and volume. This property, known as density (ρ), is essential for understanding the behavior of substances in various conditions. Unlike solids, which have definite shapes and volumes, fluids adapt to their containers, making density a crucial parameter in fluid mechanics. 🧊

What is Density?

Density is defined as the mass per unit volume of a substance:

Where:

• : Density (kg/m³ or g/cm³)

• : Mass (kg or g)

• : Volume (m³ or cm³)

Key Properties:

• Intrinsic Property: Density does not depend on the size or shape of a substance. For instance, a single balloon of helium and a thousand helium tanks will have the same density.

• Units: Density is typically expressed in kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³).

• Influence on Floating and Sinking: If an object’s density is higher than the surrounding fluid, it will sink. Conversely, objects with lower density float. 🤯

Example Problems

Example 1:

Problem: A block of metal has a mass of 50 grams and a volume of 10 cm³. What is its density?
Solution:

Example 2:

Problem: A cylinder of wood has a mass of 200 grams and a radius of 2 cm. What is its density?
Solution: First, calculate the volume of the cylinder:

Then calculate the density:

Example 3:

Problem: A swimming pool contains 50,000 liters of water with a total mass of 400,000 grams. What is the water’s density?
Solution:

Understanding Specific Gravity

Specific gravity compares an object’s density to a reference substance (usually water):

Key Points:

• Specific gravity indicates if an object will float or sink in water.

• Specific gravity > 1: The object sinks.

• Specific gravity < 1: The object floats.

Example:

Problem: A sponge has a specific gravity of 0.1 in water. What percentage of the sponge is above water?
Solution: The specific gravity (0.1) means 10% of the sponge is submerged. Therefore, 90% is above the surface. 💧

Why Density Matters

• Material Identification: Use density to compare and identify materials (e.g., gold vs. aluminum).

• Practical Applications: Understanding density helps predict whether objects will float or sink, design systems like ships and submarines, and analyze natural phenomena.

• Temperature and Pressure Effects: Density decreases with increasing temperature and increases with rising pressure.

Frequently Asked Questions

Q1: Which has greater density: 100 grams of mercury or 10,000 kilograms of mercury?
A1: Both have the same density. Density is independent of the amount of substance.

Q2: If Object A and Object B have the same density but Object B has more volume, which has more mass?
A2: Object B has more mass since mass and volume are proportional when density is constant.

Density serves as the cornerstone of fluid mechanics, offering insights into how substances interact with their environments. Mastering this concept will enhance your understanding of physical systems and prepare you for advanced problem-solving in AP Physics 2.