define and use density

⚖️ Equilibrium of Forces

In physics, equilibrium occurs when all the forces acting on an object balance each other out. Think of a seesaw perfectly level – the upward push from the fulcrum equals the downward pull of the weight on the other side.

What is Equilibrium?

• Static equilibrium: The object is at rest and all forces cancel.
• Dynamic equilibrium: The object moves at constant velocity; forces still cancel.

Mathematically, for a body at rest:

$$\sum \vec{F} = 0$$

Why Density Matters in Equilibrium

Density, denoted by ρ, tells us how much mass is packed into a given volume. It is crucial when determining buoyant forces, which often decide whether an object floats or sinks.

Definition:

$$\rho = \frac{m}{V}$$

• m = mass (kg)
• V = volume (m³)

Units: kg m⁻³ (kilograms per cubic meter).

Density in Everyday Life

Consider a wooden block and a steel block of the same size. Even though they occupy the same volume, the steel block is heavier because its density is higher. In equilibrium, the heavier block will sink in water, while the lighter one will float.

Calculating Buoyant Force

The buoyant force F_b equals the weight of the fluid displaced:

$$F_b = \rho_{\text{fluid}} \, V_{\text{displaced}} \, g$$

Where g is the acceleration due to gravity (≈9.81 m s⁻²).

Example Problem

📏 A cube of side 0.2 m is made of a material with density 1200 kg m⁻³. It is submerged in water (density 1000 kg m⁻³). Will it sink or float?

1. Calculate the cube's mass: m = ρ V = 1200 × (0.2)³ = 9.6 kg.
2. Find the buoyant force: F_b = 1000 × (0.2)³ × 9.81 ≈ 39.2 N.
3. Weight of cube: W = m g = 9.6 × 9.81 ≈ 94.2 N.
4. Since W > F_b, the cube sinks.

Quick Reference Table: Densities of Common Materials

Key Takeaways

• Equilibrium means the vector sum of all forces is zero.
• Density is mass per unit volume and is essential for predicting buoyancy.
• Use the buoyant force formula to determine if an object will float or sink.
• Remember the comparison rule: weight vs. buoyant force.