The displaced volume of a liquid is crucial in calculating buoyancy force. This volume can be determined using the formula for buoyancy force:
\[ F_b = \rho \cdot g \cdot V \]
Rearranged to solve for displaced volume:
\[ V = \frac{F_b}{\rho \cdot g} \]
Where:
- \(V\) is the displaced volume of the liquid (in cubic meters, m³)
- \(F_b\) is the buoyancy force (in newtons, N)
- \(\rho\) is the density of the liquid (in kilograms per cubic meter, kg/m³)
- \(g\) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\))
Example 1: Displaced Volume of Water by a Submerged Rock
Problem: A rock exerts a buoyancy force of \(980 \, \text{N}\) when submerged in water. The density of water is \(1000 \, \text{kg/m}^3\). What is the displaced volume of water?
Calculation:
Given:
- \(F_b = 980 \, \text{N}\)
- \(\rho = 1000 \, \text{kg/m}^3\)
- \(g = 9.8 \, \text{m/s}^2\)
Using the formula:
\[ V = \frac{F_b}{\rho \cdot g} = \frac{980}{1000 \cdot 9.8} = 0.1 \, \text{m}^3 \]
Answer: The displaced volume of water is 0.1 cubic meters.
Example 2: Displaced Volume of Oil by a Floating Object
Problem: An object floating in oil experiences a buoyancy force of \(196 \, \text{N}\). The density of the oil is \(800 \, \text{kg/m}^3\). What is the displaced volume of oil?
Calculation:
Given:
- \(F_b = 196 \, \text{N}\)
- \(\rho = 800 \, \text{kg/m}^3\)
- \(g = 9.8 \, \text{m/s}^2\)
Using the formula:
\[ V = \frac{F_b}{\rho \cdot g} = \frac{196}{800 \cdot 9.8} = 0.025 \, \text{m}^3 \]
Answer: The displaced volume of oil is 0.025 cubic meters.
Example 3: Displaced Volume of Seawater by a Submarine
Problem: A submarine experiences a buoyancy force of \(2,000,000 \, \text{N}\) when submerged in seawater. The density of seawater is \(1025 \, \text{kg/m}^3\). What is the displaced volume of seawater?
Calculation:
Given:
- \(F_b = 2,000,000 \, \text{N}\)
- \(\rho = 1025 \, \text{kg/m}^3\)
- \(g = 9.8 \, \text{m/s}^2\)
Using the formula:
\[ V = \frac{F_b}{\rho \cdot g} = \frac{2,000,000}{1025 \cdot 9.8} = 199.61 \, \text{m}^3 \]
Answer: The displaced volume of seawater is 199.61 cubic meters.
