Mechanics of Solids and Fluids
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Elasticity & Stress-Strain
Stress & Strain
Stress (σ) = F/A
Strain (ε) = ΔL/L (Longitudinal)
Strain (ε) = ΔV/V (Volume)
Strain (φ) = x/L (Shear)
Strain (ε) = ΔL/L (Longitudinal)
Strain (ε) = ΔV/V (Volume)
Strain (φ) = x/L (Shear)
Stress is restoring force per unit area. Strain is the fractional change in dimension. Both are dimensionless for strain.
Hooke's Law & Moduli
Stress ∝ Strain
Y = σₗ/εₗ = (F/A)/(ΔL/L)
B = -ΔP/(ΔV/V)
η = σₛ/φ = (F/A)/(x/L)
Y = σₗ/εₗ = (F/A)/(ΔL/L)
B = -ΔP/(ΔV/V)
η = σₛ/φ = (F/A)/(x/L)
Y: Young's Modulus, B: Bulk Modulus, η: Shear Modulus (Modulus of Rigidity).
Elastic Potential Energy
U = ½ × Stress × Strain × Volume
U = ½ × F × ΔL
Energy Density (u) = U/V
u = ½ × Y × (Strain)²
U = ½ × F × ΔL
Energy Density (u) = U/V
u = ½ × Y × (Strain)²
Energy stored in a stretched wire. Analogous to U = ½kx² for a spring.
Poisson's Ratio (σ)
σ = - (Lateral Strain) / (Longitudinal Strain)
σ = -(ΔD/D) / (ΔL/L)
σ = -(ΔD/D) / (ΔL/L)
Theoretical range: -1 < σ < 0.5. Practical range for metals: 0.2 to 0.4.
Pressure, Density & Buoyancy
Pressure & Density
Pressure (P) = F⊥/A
Density (ρ) = Mass/Volume
Relative Density = ρ_substance / ρ_water
Density (ρ) = Mass/Volume
Relative Density = ρ_substance / ρ_water
SI unit of Pressure is Pascal (Pa) or N/m². 1 atm ≈ 1.013 × 10⁵ Pa.
Pressure Variation with Depth
P = P₀ + hρg
ΔP = hρg
P_gauge = P_absolute - P_atm
ΔP = hρg
P_gauge = P_absolute - P_atm
P₀ is the pressure at the surface (usually atmospheric pressure).
Pascal's Law
P₁ = P₂
F₁/A₁ = F₂/A₂
F₁/A₁ = F₂/A₂
Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid. Basis of hydraulic lifts.
Archimedes' Principle
Buoyant Force (F₈) = V_submerged × ρ_fluid × g
F₈ = Weight of displaced fluid
Apparent Weight = W_actual - F₈
F₈ = Weight of displaced fluid
Apparent Weight = W_actual - F₈
For a floating body, its weight is equal to the buoyant force.
Fluid Dynamics
Equation of Continuity
A₁v₁ = A₂v₂
Av = Constant
Av = Constant
Based on the conservation of mass for an incompressible, non-viscous fluid. Where area is smaller, speed is greater.
Bernoulli's Theorem
P + ½ρv² + ρgh = Constant
P/ρg + v²/2g + h = Constant
P/ρg + v²/2g + h = Constant
Based on the conservation of energy for an ideal fluid. P is static pressure, ½ρv² is dynamic pressure, ρgh is hydrostatic pressure.
Torricelli's Law (Efflux)
Velocity of Efflux
v = √(2gh)
v = √(2gh)
A direct application of Bernoulli's theorem. 'h' is the height of the fluid surface above the orifice. Speed is same as a freely falling body from height h.
Viscosity & Surface Tension
Surface Tension (T)
T = F/L
Surface Energy (U) = T × ΔA
Surface Energy (U) = T × ΔA
Force per unit length acting perpendicular to an imaginary line on the liquid surface. Tends to minimize surface area.
Excess Pressure
Liquid Drop: ΔP = 2T/R
Soap Bubble: ΔP = 4T/R
Air Bubble in Liquid: ΔP = 2T/R
Soap Bubble: ΔP = 4T/R
Air Bubble in Liquid: ΔP = 2T/R
A soap bubble has two free surfaces (inner and outer), hence the factor of 4.
Capillary Action (Rise/Fall)
h = (2T cosθ) / (rρg)
h ∝ 1/r
h ∝ 1/r
θ is the angle of contact. For water in glass, θ < 90° (rise). For mercury in glass, θ > 90° (fall).
Stokes' Law & Terminal Velocity
Viscous Force: Fᵥ = 6πηrv
Terminal Velocity (vₜ):
vₜ = [2r²(ρ - σ)g] / 9η
Terminal Velocity (vₜ):
vₜ = [2r²(ρ - σ)g] / 9η
ρ is density of object, σ is density of fluid. At vₜ, Net Force = 0 (Weight = Fᵥ + F₈).
BITSAT Hot-Tips
Speed & Accuracy Hacks
**Dimensional Analysis:** Before solving, check dimensions of options. Stress, Pressure, and Moduli all have the same dimension [ML⁻¹T⁻²]. This can instantly eliminate wrong choices.
**Bernoulli's is Energy Conservation:** Think of P as potential energy density, ½ρv² as kinetic energy density, and ρgh as gravitational potential energy density. Their sum is constant.
**Apparent Weight Shortcut:** Apparent weight W' = W(1 - ρ_fluid / ρ_body). If ρ_fluid > ρ_body, the object floats (W' is negative).
**Terminal Velocity Proportionality:** For questions asking for ratios, remember vₜ ∝ r². If radius is doubled, terminal velocity becomes 4 times.
**Continuity Check:** In any pipe flow problem, first apply A₁v₁ = A₂v₂. It's the simplest relation and often gives you one variable directly.
**Bubble Trouble:** Always double-check if it's a 'liquid drop' (2T/R) or a 'soap bubble' (4T/R). BITSAT loves to trick students here.
Key Concepts at a Glance
| Concept | Governing Principle | BITSAT Application |
|---|---|---|
Elasticity | Restoring force per unit area is proportional to fractional change in dimension. | Calculate wire extension, energy stored, or identify material based on Modulus. |
Buoyancy | Upward force equals the weight of the fluid displaced (Archimedes'). | Problems on floating/sinking, apparent weight, and fraction of volume submerged. |
Fluid Flow | Conservation of Mass (Continuity) & Energy (Bernoulli's). | Find fluid speed/pressure in pipes, venturi-meter, airplane lift, efflux velocity. |
Surface Tension | Liquid surfaces behave like a stretched membrane due to cohesive forces. | Capillary rise, excess pressure in bubbles, work done to form a drop/film. |
Viscosity | Internal friction in a fluid resisting relative motion between layers. | Terminal velocity of falling objects in fluids (rain drops, ball bearings). |