Physics by Lamhi: Not Your Boring Physics

CLASS 11 · CHAPTER 8 · MATTER

QUICK REVISION

Mechanical Properties of Solids

Every key result from this chapter, boxed and ready for a last look before the exam. No derivations here, just what to recall and when to use it — for the full explanation, see the detailed notes.

1. Stress & Strain — the Three Kinds

stress=restoring forcearea\text{stress} = \dfrac{\text{restoring force}}{\text{area}}

Units Pa (N/m²), dimensions [ML−1T−2][ML^{-1}T^{-2}]. Strain is a ratio of lengths/volumes — no units, no dimensions.

  1. Longitudinal (tensile/compressive): force along the length; strain =ΔL/L= \Delta L/L.
  2. Shearing: tangential force on one face, opposite face fixed; strain =θ≈Δx/L= \theta \approx \Delta x/L, no volume change.
  3. Volume (hydraulic): uniform inward pressure; stress =ΔP= \Delta P, strain =ΔV/V= \Delta V/V.

2. Hooke's Law

Hooke's law
stress=k×strain\text{stress} = k \times \text{strain}

Holds only within the elastic limit; kk is whichever modulus (Y, G, or B) matches the kind of stress/strain in play.

3. The Stress-Strain Curve — Region by Region

  1. O to A (elastic limit/yield point): Hooke’s law holds, fully elastic — loading and unloading retrace the same line, nothing left behind.
  2. Past A (plastic flow): strain grows faster than stress; release the load here and unloading follows a NEW line parallel to OA (same slope YY) — leaves a permanent set.
  3. At B (ultimate tensile strength): maximum stress the material can bear; beyond B the wire keeps stretching (necking) while the stress it sustains drops.
  4. At E (fracture point): the wire snaps.
  5. Ductile vs brittle: large plastic region (A to E) → ductile (most metals); little/none → brittle (glass, cast iron, ceramics).
permanent set=εmax reached−stress at that pointY\text{permanent set} = \varepsilon_{\text{max reached}} - \dfrac{\text{stress at that point}}{Y}

Depends on the highest strain ever reached, not the current one — the wire “remembers” its worst stretch.

4. The Three Moduli

Young's modulus
Y=longitudinal stresslongitudinal strain=F/AΔL/LY = \dfrac{\text{longitudinal stress}}{\text{longitudinal strain}} = \dfrac{F/A}{\Delta L/L}
ΔL=FLAY\Delta L = \dfrac{FL}{AY}
Shear modulus
G=shearing stressshearing strain=F/AΔx/L=F/AθG = \dfrac{\text{shearing stress}}{\text{shearing strain}} = \dfrac{F/A}{\Delta x/L} = \dfrac{F/A}{\theta}
Bulk modulus
B=−ΔPΔV/VB = -\dfrac{\Delta P}{\Delta V/V}

Compressibility =1/B= 1/B. Steel Y≈2×1011 PaY \approx 2\times10^{11}\text{ Pa} (stiff — cables, rails, beams); rubber Y≈105–106 PaY \approx 10^{5}\text{–}10^{6}\text{ Pa} (large reversible deformation). GG is always smaller than YY for the same material, typically by a factor of 2–3.

5. Poisson's Ratio & the Y–G–σ Link

σ=−lateral strainlongitudinal strain\sigma = -\dfrac{\text{lateral strain}}{\text{longitudinal strain}}

Typically 0.2–0.4 (~0.3 for metals); theory restricts −1<σ<0.5-1 < \sigma < 0.5, with σ=0.5\sigma = 0.5 meaning constant volume while stretching (rubber, nearly).

Y=2G(1+σ)Y = 2G(1+\sigma)

Know any two of YY, GG, σ\sigma and the third is fixed.

6. Elastic Potential Energy

Elastic potential energy
U=12F ΔL=12×stress×strain×volumeU = \dfrac{1}{2}F\,\Delta L = \dfrac{1}{2}\times\text{stress}\times\text{strain}\times\text{volume}

The second form is energy stored per unit volume — applies alike to a stretched wire, a twisted shaft, or a compressed block.

7. Design Notes — Factor of Safety

  1. Cables/beams are operated well inside the elastic region (a factor of safety), never near the ultimate tensile strength.
  2. Loading past the elastic limit even once leaves a permanent set and weakens the material; repeating it causes fatigue failure.
  3. Metal beats wood for beams: larger YY (less sag) + higher elastic limit (more load before permanent bending).
  4. Hollow box / I-section beams concentrate material at the top and bottom edges, where bending stress is largest — more resistance for less weight.

Full derivations and worked examples: detailed notes →