define and use density

Equilibrium of Forces

Learning Objective

Define density and use it to solve equilibrium problems involving forces.

1. What is Density?

Density ($\rho$) is the mass per unit volume of a material.

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

• $\rho$ – density (kg m⁻³)
• $m$ – mass (kg)
• $V$ – volume (m³)

Re‑arranging the formula allows us to find mass or volume when the other quantities are known:

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

2. Units and Typical \cdot alues

Material	Density (kg m⁻³)
Water (4 °C)	1000
Aluminium	2700
Steel	7850
Air (STP)	1.29

3. Equilibrium of Forces

A body is in static equilibrium when both the net force and the net torque acting on it are zero:

$$\sum \mathbf{F} = \mathbf{0},\qquad \sum \boldsymbol{\tau} = \mathbf{0}$$

• Translational equilibrium → no linear acceleration.
• Rotational equilibrium → no angular acceleration.

4. Using Density in Equilibrium Problems

Density often appears when dealing with forces that depend on mass or volume, such as weight, buoyancy, and pressure.

4.1 Weight

The weight $W$ of an object is the force due to gravity:

$$W = mg = \rho \cdot g$$

where $g = 9.81\ \text{m s}^{-2}$.

4.2 Buoyant Force

When an object is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the displaced fluid:

$$F_{\text{b}} = \rho_{\text{fluid}} V_{\text{disp}} g$$

This principle is used to determine whether an object will float, sink, or remain suspended.

4.3 Pressure on Surfaces

Pressure $P$ is force per unit area. For a fluid at depth $h$, the pressure is:

$$P = \rho_{\text{fluid}} g h$$

The resultant force on a flat surface of area $A$ is $F = PA$.

5. Worked Example

1. Calculate the mass of a solid aluminium cube with side length $0.10\ \text{m}$.
2. Determine whether the cube will float in water.

Solution:

1. Volume of the cube:

$$V = (0.10\ \text{m})^{3} = 1.0\times10^{-3}\ \text{m}^{3}$$

Mass using $\rho_{\text{Al}} = 2700\ \text{kg m}^{-3}$:

$$m = \rho V = 2700 \times 1.0\times10^{-3} = 2.7\ \text{kg}$$

2. Weight of the cube:

$$W = mg = 2.7 \times 9.81 = 26.5\ \text{N}$$

Buoyant force in water ($\rho_{\text{water}} = 1000\ \text{kg m}^{-3}$):

$$F_{\text{b}} = \rho_{\text{water}} V g = 1000 \times 1.0\times10^{-3} \times 9.81 = 9.81\ \text{N}$$

Since $W > F_{\text{b}}$, the cube sinks. For equilibrium (floating), the condition $W = F_{\text{b}}$ would require a material density equal to that of water.

6. Summary Checklist

• Density definition: $\rho = m/V$.
• Weight can be written as $W = \rho \cdot g$.
• Buoyant force: $F_{\text{b}} = \rho_{\text{fluid}} V_{\text{disp}} g$.
• Static equilibrium: $\sum \mathbf{F}=0$ and $\sum \boldsymbol{\tau}=0$.
• Use density to convert between mass and volume when calculating forces.