Properties of Solids and Liquids Formula Sheet

Every must-know formula, constant, and result for elasticity and fluid mechanics — compiled for last-minute JEE revision. Each section maps to a full topic page; follow the links if a result needs derivation.

Elasticity: Stress and Strain

Stress — internal restoring force per unit area; same dimensions as pressure, $[ML^{-1}T^{-2}]$, unit Pa.

$$\boxed{\text{Stress} = \frac{F}{A}}$$

Strain — relative deformation; dimensionless (a pure ratio).

$$\boxed{\text{Strain} = \frac{\text{Change in dimension}}{\text{Original dimension}}}$$

Type	Stress	Strain
Tensile / Compressive	$F/A$ (longitudinal)	$\dfrac{\Delta L}{L}$
Shearing (tangential)	$F_{\text{tangential}}/A$	$\theta = \dfrac{\Delta x}{L}$
Hydraulic (bulk)	$-\Delta P$	$\dfrac{\Delta V}{V}$

Hooke’s Law (within elastic limit):

$$\boxed{\text{Stress} = E \times \text{Strain}}$$

For a spring: $\boxed{F = kx}$

Elastic Potential Energy

$$\boxed{U = \tfrac{1}{2}kx^2}$$

For a stretched wire / rod:

$$\boxed{U = \tfrac{1}{2}\,\text{Stress}\times\text{Strain}\times\text{Volume} = \tfrac{1}{2}F\,\Delta L}$$

Energy density (per unit volume):

$$\boxed{u = \frac{U}{V} = \tfrac{1}{2}\,\text{Stress}\times\text{Strain}}$$

Full page: Stress and Strain

Elastic Moduli

$$\boxed{\text{Elastic Modulus} = \frac{\text{Stress}}{\text{Strain}}}$$

Common unit: Pa; dimensions $[ML^{-1}T^{-2}]$.

Modulus	Formula	Applies to
Young’s $Y$	$\dfrac{F/A}{\Delta L/L} = \dfrac{FL}{A\,\Delta L}$	Solids
Bulk $K$	$\dfrac{-\Delta P}{\Delta V/V} = -V\dfrac{\Delta P}{\Delta V}$	Solids, liquids, gases
Shear (rigidity) $\eta$	$\dfrac{F/A}{\theta} = \dfrac{F/A}{\Delta x/L}$	Solids only
Poisson’s ratio $\sigma$	$-\dfrac{\Delta d/d}{\Delta L/L}$	Solids (dimensionless)
Compressibility $\kappa$	$\dfrac{1}{K}$	All materials

For a wire, $\Delta L = \dfrac{FL}{AY}$, so it behaves like a spring with $k = \dfrac{YA}{L}$.

Bulk modulus of an ideal gas:

$$\boxed{K_{\text{isothermal}} = P} \qquad \boxed{K_{\text{adiabatic}} = \gamma P}$$

where $\gamma = C_p/C_v$ (air: $\gamma \approx 1.4$).

Poisson’s ratio range: $0 < \sigma < 0.5$ (cork $\approx 0$, rubber $\approx 0.5$, most metals $\approx 0.2$–$0.4$).

Relations Between Elastic Constants (isotropic materials)

$$\boxed{Y = 2\eta(1+\sigma)} \qquad \boxed{Y = 3K(1-2\sigma)}$$$$\boxed{K = \frac{Y}{3(1-2\sigma)}} \qquad \boxed{\eta = \frac{Y}{2(1+\sigma)}}$$$$\boxed{K = \frac{\eta(2+2\sigma)}{3(1-2\sigma)}} \qquad \boxed{Y = \frac{9K\eta}{3K+\eta}}$$

Reference values (Young’s modulus): Steel $2\times10^{11}$ Pa, Iron $1.9\times10^{11}$, Copper $1.2\times10^{11}$, Aluminium $0.7\times10^{11}$, Rubber $\approx 5\times10^6$ Pa.

Full page: Elastic Moduli

Fluid Pressure and Pascal’s Law

$$\boxed{P = \frac{F}{A}}$$

Pressure in a fluid at rest is a scalar — same in all directions at a point.

Hydrostatic pressure at depth $h$:

$$\boxed{P = P_0 + \rho g h}$$

Gauge pressure:

$$\boxed{P_{\text{gauge}} = P - P_0 = \rho g h}$$

Pascal’s Law — hydraulic lift:

$$\boxed{\frac{F_1}{A_1} = \frac{F_2}{A_2}} \qquad \frac{F_2}{F_1} = \frac{A_2}{A_1}$$

Work conserved: $F_1 d_1 = F_2 d_2$.

Barometer / manometer: $P_0 = \rho_{Hg}\, g\, h$; open-tube manometer gives $P_{\text{gas}} - P_0 = \rho g h$.

Archimedes’ Principle and Buoyancy

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

Floating condition ($\rho_{\text{body}} < \rho_{\text{fluid}}$):

$$\boxed{\frac{V_{\text{submerged}}}{V_{\text{total}}} = \frac{\rho_{\text{body}}}{\rho_{\text{fluid}}}}$$

Apparent weight (sinking body):

$$\boxed{W_{\text{apparent}} = W_{\text{actual}} - F_B = Vg(\rho_{\text{body}} - \rho_{\text{fluid}})}$$

Full page: Fluid Pressure and Pascal’s Law

Fluid Dynamics: Continuity and Bernoulli

Equation of continuity (incompressible, streamline flow):

$$\boxed{A_1 v_1 = A_2 v_2 = \text{constant}}$$

Bernoulli’s theorem (ideal fluid — non-viscous, incompressible, steady, streamline flow):

$$\boxed{P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}}$$

Equivalently, dividing by $\rho g$ (head form):

$$\boxed{\frac{P}{\rho g} + \frac{v^2}{2g} + h = \text{constant}}$$

Key Applications

Application	Formula	Condition
Torricelli’s efflux	$v = \sqrt{2gh}$	Hole at depth $h$, large tank
Venturi meter	$v_1 = A_2\sqrt{\dfrac{2(P_1-P_2)}{\rho(A_1^2 - A_2^2)}}$	Horizontal pipe
Flow rate	$Q = A_1 v_1 = Av$	—
Aerofoil lift	$F_{\text{lift}} = \tfrac{1}{2}\rho(v_{\text{top}}^2 - v_{\text{bottom}}^2)A$	Wing area $A$

$$\boxed{v = \sqrt{2gh}} \qquad \text{(Torricelli — same as free fall from height } h)$$

Full page: Bernoulli’s Theorem

Viscosity, Stokes’ Law and Flow Through Pipes

Newton’s law of viscosity:

$$\boxed{F = -\eta A \frac{dv}{dy}}$$

$\eta$ = coefficient of viscosity. Unit: Pa·s (CGS: poise; $1$ poise $= 0.1$ Pa·s, $1$ cP $= 10^{-3}$ Pa·s). Dimensions $[ML^{-1}T^{-1}]$. Water at 20 °C: $\eta \approx 10^{-3}$ Pa·s.

Reynolds number (dimensionless):

$$\boxed{Re = \frac{\rho v D}{\eta}}$$

Laminar: $Re < 1000$; turbulent: $Re > 2000$; transition: $1000 < Re < 2000$.

Stokes’ law (small sphere, low velocity, $Re < 1$):

$$\boxed{F = 6\pi \eta r v}$$

Terminal velocity:

$$\boxed{v_t = \frac{2r^2 g(\rho_{\text{sphere}} - \rho_{\text{fluid}})}{9\eta}}$$

When $\rho_{\text{sphere}} \gg \rho_{\text{fluid}}$: $v_t = \dfrac{2r^2 \rho g}{9\eta}$.

Time to reach $\sim$90 % of $v_t$: $t_{90\%} \approx \dfrac{m}{3\pi\eta r}$.

Poiseuille’s law (laminar flow through a cylindrical pipe):

$$\boxed{Q = \frac{\pi r^4 (P_1 - P_2)}{8\eta L}}$$

Full page: Viscosity and Stokes’ Law

Surface Tension and Capillarity

Two equivalent definitions, same value and units ($\text{N/m} = \text{J/m}^2$), dimensions $[MT^{-2}]$:

$$\boxed{\gamma = \frac{F}{L}} \qquad \boxed{\gamma = \frac{W}{A}}$$

Work to create new surface (surface energy):

$$\boxed{W = \gamma\,\Delta A} \qquad U = \gamma A$$

Excess Pressure

Geometry	Excess pressure	Surfaces
Liquid drop	$\dfrac{2\gamma}{r}$	1
Air bubble in liquid	$\dfrac{2\gamma}{r}$	1
Soap bubble	$\dfrac{4\gamma}{r}$	2 (inner + outer)

$$\boxed{\Delta P_{\text{drop}} = \frac{2\gamma}{r}} \qquad \boxed{\Delta P_{\text{soap bubble}} = \frac{4\gamma}{r}}$$

Capillary Rise

$$\boxed{h = \frac{2\gamma\cos\theta}{\rho g r}}$$

For perfect wetting ($\theta = 0°$, $\cos\theta = 1$): $h = \dfrac{2\gamma}{\rho g r}$.

Obtuse $\theta$ (e.g. mercury in glass) gives $\cos\theta < 0$ → capillary depression.

Angle of contact: measured through the liquid. $\theta < 90°$ → wets, concave meniscus, adhesion > cohesion (water on glass). $\theta > 90°$ → doesn’t wet, convex meniscus, cohesion > adhesion (mercury on glass, $\theta \approx 135°$–$140°$).

Full page: Surface Tension and Capillarity

Coalescing Drops and Bubbles (Quick Results)

graph LR
    A["n small drops/bubbles
radius r"] --> B{Combine}
    B --> C["Liquid drops:
volume conserved
R = n^(1/3) r"]
    B --> D["Soap bubbles, isothermal:
surface energy/area conserved
R² = r₁² + r₂² + ..."]

• Liquid drops coalescing — volume is conserved: $R^3 = \sum r_i^3$ (e.g. $n=8 \Rightarrow R = 2r$).
• Soap bubbles coalescing (isothermal) — surface energy is conserved: $R^2 = r_1^2 + r_2^2$.

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One-Glance Master Table

Quantity	Formula	Notes
Stress	$\sigma = F/A$	Pa, $[ML^{-1}T^{-2}]$
Strain	$\varepsilon = \Delta L/L$	dimensionless
Hooke’s law	$\sigma = E\varepsilon$; $F = kx$	within elastic limit
Elastic PE	$U = \tfrac12 kx^2 = \tfrac12 F\Delta L$	—
Young’s modulus	$Y = FL/(A\Delta L)$	solids
Bulk modulus	$K = -V\Delta P/\Delta V$	all states; gas: $K_{\text{iso}}=P$, $K_{\text{adi}}=\gamma P$
Shear modulus	$\eta = (F/A)/\theta$	solids only
Poisson’s ratio	$\sigma = -(\Delta d/d)/(\Delta L/L)$	$0$–$0.5$
Modulus relation	$Y = 9K\eta/(3K+\eta)$	isotropic
Hydrostatic pressure	$P = P_0 + \rho gh$	depends only on $h$
Hydraulic lift	$F_1/A_1 = F_2/A_2$	$F_1 d_1 = F_2 d_2$
Buoyant force	$F_B = \rho_{\text{fluid}} V_{\text{disp}}\, g$	upward
Floating fraction	$V_{\text{sub}}/V = \rho_{\text{body}}/\rho_{\text{fluid}}$	$\rho_{\text{body}}<\rho_{\text{fluid}}$
Continuity	$Av = $ const	incompressible
Bernoulli	$P + \tfrac12\rho v^2 + \rho gh = $ const	ideal, same streamline
Torricelli	$v = \sqrt{2gh}$	efflux
Venturi	$v_1 = A_2\sqrt{2\Delta P/[\rho(A_1^2-A_2^2)]}$	horizontal
Newton’s viscosity	$F = -\eta A\, dv/dy$	$\eta$ in Pa·s
Reynolds number	$Re = \rho v D/\eta$	$<1000$ laminar, $>2000$ turbulent
Stokes’ law	$F = 6\pi\eta r v$	small sphere, $Re<1$
Terminal velocity	$v_t = 2r^2 g(\rho_s-\rho_f)/(9\eta)$	$\propto r^2$
Poiseuille	$Q = \pi r^4 \Delta P/(8\eta L)$	$Q\propto r^4$
Surface tension	$\gamma = F/L = W/A$	N/m $=$ J/m²
Excess pressure (drop)	$\Delta P = 2\gamma/r$	one surface
Excess pressure (bubble)	$\Delta P = 4\gamma/r$	two surfaces
Capillary rise	$h = 2\gamma\cos\theta/(\rho gr)$	narrow tubes

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Related Pages

• Stress and Strain
• Elastic Moduli
• Fluid Pressure and Pascal’s Law
• Bernoulli’s Theorem
• Viscosity and Stokes’ Law
• Surface Tension and Capillarity