Section 1.2: Density
This section introduces density, which is the mass per unit volume of a substance. Understanding density is crucial for fluid mechanics, buoyancy, and material identification.
Key Equations & Concepts:
- Density: \( \rho = \frac{m}{V} \)
- Units: kg/m³ (SI), g/cm³ (common)
- Relationship with Buoyant Force: Denser fluids exert more buoyant force
- Specific Gravity: \( SG = \frac{\rho_{object}}{\rho_{water}} \)
Example 1
A metal block has mass 12 kg and volume 0.004 m³. Find its density.
\( \rho = \frac{m}{V} = \frac{12}{0.004} = 3000 \text{ kg/m³} \)
Example 2
An object floats in water with 60% of its volume submerged. If the density of water is 1000 kg/m³, find the density of the object.
\( \rho_{object} = \text{fraction submerged} \times \rho_{fluid} = 0.6 \cdot 1000 = 600 \text{ kg/m³} \)
Practice Problems
- Find the density of a liquid with mass 2 kg and volume 2.5 L.
- A cube of side 0.1 m weighs 8 kg. Calculate its density.
- Specific gravity of an object is 0.85. Find its density.
- A 5 kg object floats in water, 40% submerged. Find its volume.
- Density of mercury is 13,600 kg/m³. Find the mass of 0.02 m³ of mercury.
- An object of density 700 kg/m³ is placed in water. Will it float or sink?
- A material weighs 90 N in air, 60 N in water. Find its density.
- A cylinder has mass 3 kg, radius 0.05 m, height 0.2 m. Compute its density.
- Find the volume of 10 kg of oil if \( \rho = 900 \text{ kg/m³} \).
- An object floats in a liquid, 75% submerged. If \( \rho_{liquid} = 1000 \text{ kg/m³} \), find object’s density.