PiTheory

Mechanical Properties of Solids formulas

Class 11 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 8, the special cases questions are built on, and diagrams where they help.

21 formulas5 sectionsClass 11 · Chapter 82 of 5 sections free

By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE

Most used formulasOther formulas and cases

1Stress, strain and moduli

$$\text{stress}=\frac FA,\qquad Y=\frac{F/A}{\Delta L/L}=\frac{FL}{A\,\Delta L},\qquad \Delta L=\frac{FL}{AY}=\frac{4FL}{\pi d^2Y}$$
$$\begin{array}{l}\displaystyle \text{stress}=\frac FA\\[6pt]\displaystyle Y=\frac{F/A}{\Delta L/L}=\frac{FL}{A\,\Delta L}\\[6pt]\displaystyle \Delta L=\frac{FL}{AY}=\frac{4FL}{\pi d^2Y}\end{array}$$

Hooke's law: stress ∝ strain within the elastic limit. Stress and moduli in N m⁻² (Pa); strain has no unit. $Y$ depends only on the material (and temperature: $Y$ falls when heated), not on $L$ or $A$.

$$\begin{array}{l}\displaystyle B=-\frac{\Delta P}{\Delta V/V},\quad \text{compressibility}=\frac1B\\[5pt]\displaystyle G=\eta=\frac{F/A}{\theta}\ (\text{shear})\end{array}$$
$$\begin{array}{l}\displaystyle B=-\frac{\Delta P}{\Delta V/V}\\[6pt]\displaystyle \text{compressibility}=\frac1B\\[6pt]\displaystyle G=\eta=\frac{F/A}{\theta}\ (\text{shear})\end{array}$$

Solids have $Y,B,G$; liquids and gases only $B$. Gas: $B_{\text{iso}}=P$, $B_{\text{ad}}=\gamma P$. Water at depth $h$: $\dfrac{\Delta V}V=\dfrac{\rho gh}B$, density rises to $\rho\left(1+\dfrac{\rho gh}B\right)$. Shear: $\theta=\dfrac{x}{L}$ (lateral shift / height).

strainstressABCDEHooke: slope YA proportional limitB elastic limit / yieldD ultimate (tensile) strengthE fracturedashed: permanent setDE long: ductile; short: brittle
  • Steel is more elastic than rubber (larger $Y$). Quartz: nearly perfectly elastic; putty: plastic. Ductile (copper, aluminium): large plastic region; brittle (glass, cast iron): breaks soon after elastic limit. Elastomers (rubber, aorta): no clear plastic region, large strain.
  • Elastic fatigue and elastic after-effect; hysteresis in rubber (area of loop = heat lost per unit volume).

2Poisson's ratio and relations

$$\sigma=-\frac{\Delta d/d}{\Delta L/L},\qquad \frac{\Delta V}V=(1-2\sigma)\frac{\Delta L}L,\qquad \frac{\Delta A}A=-2\sigma\frac{\Delta L}L$$
$$\begin{array}{l}\displaystyle \sigma=-\frac{\Delta d/d}{\Delta L/L}\\[6pt]\displaystyle \frac{\Delta V}V=(1-2\sigma)\frac{\Delta L}L\\[6pt]\displaystyle \frac{\Delta A}A=-2\sigma\frac{\Delta L}L\end{array}$$

Theoretical range $-1<\sigma<\tfrac12$; practically $0$ to $0.5$. $\sigma=0.5$: no volume change (rubber ≈0.49).

$$\begin{array}{l}\displaystyle Y=3B(1-2\sigma)=2G(1+\sigma)\\[5pt]\displaystyle \frac9Y=\frac1B+\frac3G,\quad \sigma=\frac{3B-2G}{6B+2G}\end{array}$$
$$\begin{array}{l}\displaystyle Y=3B(1-2\sigma)=2G(1+\sigma)\\[6pt]\displaystyle \frac9Y=\frac1B+\frac3G\\[6pt]\displaystyle \sigma=\frac{3B-2G}{6B+2G}\end{array}$$

3 more sections and 9 formulas in the full chapter

  1. 3Elongation cases2 formulas · 1 case table
  2. 4Elastic energy2 formulas
  3. 5Bending and torsion5 formulas

Get the complete Mechanical Properties of Solids sheet, with the whole Class 11 Physics Formula Book (all chapters, as a PDF), free on WhatsApp.

Practise these formulas27 Mechanical Properties of Solids questions with step-by-step solutions →