Density is a fundamental property of liquids that affects buoyancy and other physical behaviors. It can be determined using the formula for buoyancy force:
\[ F_b = \rho \cdot g \cdot V \]
Rearranged to solve for density:
\[ \rho = \frac{F_b}{g \cdot V} \]
Where:
- \(\rho\) is the density of the liquid (in kilograms per cubic meter, kg/m³)
- \(F_b\) is the buoyancy force (in newtons, N)
- \(g\) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\))
- \(V\) is the displaced volume of the liquid (in cubic meters, m³)
Example 1: Density of a Liquid in a Container
Problem: A container filled with a liquid exerts a buoyancy force of \(196 \, \text{N}\) on an object with a volume of \(0.02 \, \text{m}^3\). What is the density of the liquid?
Calculation:
Given:
- \(F_b = 196 \, \text{N}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(V = 0.02 \, \text{m}^3\)
Using the formula:
\[ \rho = \frac{F_b}{g \cdot V} = \frac{196}{9.8 \cdot 0.02} = 1000 \, \text{kg/m}^3 \]
Answer: The density of the liquid is 1000 kg/m³.
Example 2: Density of Oil
Problem: An oil tank exerts a buoyancy force of \(490 \, \text{N}\) on a block with a volume of \(0.1 \, \text{m}^3\). What is the density of the oil?
Calculation:
Given:
- \(F_b = 490 \, \text{N}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(V = 0.1 \, \text{m}^3\)
Using the formula:
\[ \rho = \frac{F_b}{g \cdot V} = \frac{490}{9.8 \cdot 0.1} = 500 \, \text{kg/m}^3 \]
Answer: The density of the oil is 500 kg/m³.
Example 3: Density of a Mystery Liquid
Problem: A mystery liquid exerts a buoyancy force of \(2940 \, \text{N}\) on a solid object with a volume of \(0.3 \, \text{m}^3\). What is the density of the mystery liquid?
Calculation:
Given:
- \(F_b = 2940 \, \text{N}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(V = 0.3 \, \text{m}^3\)
Using the formula:
\[ \rho = \frac{F_b}{g \cdot V} = \frac{2940}{9.8 \cdot 0.3} = 1000 \, \text{kg/m}^3 \]
Answer: The density of the mystery liquid is 1000 kg/m³.
