Comprehensive theory, key formulas, diagrams, and memory aids for Mechanical Properties of Matter.
Density ($\rho$) is defined as mass per unit volume: $$\rho = \frac{m}{V}$$
It is a scalar quantity measured in kg m⁻³. Density is an intrinsic property of a material, independent of the amount of material present. For comparison: water ≈ 1000 kg m⁻³, aluminium ≈ 2700 kg m⁻³, gold ≈ 19,300 kg m⁻³, air ≈ 1.2 kg m⁻³.
Upthrust and Archimedes' Principle: A fluid exerts an upward force (upthrust) on a submerged object equal to the weight of fluid displaced: $U = \rho_{fluid} V_{submerged} g$. Objects float when upthrust equals weight.
Hooke's Law states that the extension (or compression) of a spring is directly proportional to the applied force, provided the elastic limit is not exceeded: $$F = kx$$
Where: - $F$ = applied force (N) - $k$ = spring constant, or stiffness (N m⁻¹) - $x$ = extension from natural length (m)
A higher spring constant means a stiffer spring requiring more force for the same extension. The spring constant can be determined from the gradient of a force–extension graph in the linear (Hooke's Law) region.
Springs in series: $\frac{1}{k_{total}} = \frac{1}{k_1} + \frac{1}{k_2}$ (effective constant is less than either individual spring — system is more flexible)
Springs in parallel: $k_{total} = k_1 + k_2$ (system is stiffer)
As force on a material increases beyond the elastic limit, its behaviour changes:
For a ductile material (e.g., copper, mild steel): 1. Proportionality limit (P): Hooke's Law holds — linear region. 2. Elastic limit (E): Beyond this, permanent deformation occurs. 3. Yield point (Y): Material suddenly extends significantly under approximately constant stress. Upper and lower yield points are observed in mild steel. 4. Ultimate Tensile Stress (UTS): Maximum stress the material can withstand. 5. Fracture point (F): Material breaks.
Brittle materials (glass, ceramic): Fracture without significant plastic deformation. No yield point — fractures suddenly from the proportional limit region.
Polymers (rubber): Show a complex non-linear relationship with significant plastic deformation. Rubber bands show hysteresis — the loading and unloading curves are different, with the area between them representing thermal energy dissipated.
When a force is applied to a material, we describe the deformation using:
Tensile stress ($\sigma$): Force per unit cross-sectional area: $$\sigma = \frac{F}{A}$$ Unit: Pa (N m⁻²). Stress accounts for the size of the material.
Tensile strain ($\varepsilon$): Fractional extension — the ratio of extension to original length: $$\varepsilon = \frac{x}{L_0}$$ Dimensionless. Note: strain has no unit, but can be expressed as a percentage.
Young's Modulus ($E$): A measure of the stiffness of a material — the ratio of tensile stress to tensile strain within the elastic (proportional) region: $$E = \frac{\sigma}{\varepsilon} = \frac{F/A}{x/L_0} = \frac{FL_0}{Ax}$$ Unit: Pa (N m⁻²). Young's modulus is an intrinsic material property — it does not depend on the dimensions of the sample.
| Material | Young's Modulus (GPa) |
|---|---|
| Steel | 210 |
| Aluminium | 70 |
| Copper | 130 |
| Glass | 70 |
| Rubber | 0.01–0.1 |
Young's modulus = gradient of the linear portion of the stress–strain graph.
graph TD
A[Force Applied to Material] --> B[Stress = F/A]
A --> C[Strain = x/L₀]
B --> D[Young's Modulus E = σ/ε]
C --> D
D --> E{Within Elastic Limit?}
E -- Yes --> F[Material returns to original shape]
E -- No --> G[Permanent plastic deformation]
Within the elastic limit, the work done stretching a spring or material is stored as elastic potential energy, equal to the area of the triangle under the force-extension graph: $$E_{elastic} = \frac{1}{2}Fx = \frac{1}{2}kx^2 = \frac{F^2}{2k}$$
For a material (not just a spring): $E_{elastic} = \frac{1}{2}\sigma\varepsilon \times (\text{volume of material})$
The area under the stress–strain graph equals the elastic potential energy stored per unit volume (energy density).
Strength refers to the ability to withstand stress without breaking. The ultimate tensile stress (UTS) or breaking stress is the maximum stress a material can endure.
Method using a long wire: 1. Hang a long, uniform wire vertically with a known load. 2. Measure original length $L_0$ and diameter $d$ (to find cross-section $A = \pi d^2/4$). 3. Apply incremental masses $m$; record extension $x$ for each. 4. Plot force vs extension: gradient = $k = F/x$. 5. Young's modulus: $E = \frac{F L_0}{A x} = \frac{k L_0}{A}$
Using a long, thin wire minimises the percentage uncertainty in measuring small extensions and provides a large $L_0/A$ ratio, amplifying the measurable extension.