Physics Properties of Solids and Liquids JEE Syllabus

Why does a rubber band regain its shape while clay remains deformed? Why do ships float and water droplets appear spherical? Properties of Solids and Liquids explains these everyday phenomena through concepts such as elasticity, pressure, buoyancy, viscosity, and surface tension.

Understanding the syllabus helps you see how these topics are connected and provides a clear roadmap for preparation. A strong grasp of the chapter's key concepts can make revision more effective and help you approach JEE questions with greater confidence.

Introduction to Elastic Behavior and Hooke's Law

Elasticity is the property of a material by virtue of which it regains its original shape and size after the removal of deforming forces. It highlights how internal molecular bonds counteract external distorting stress profiles.

Deforming Mechanisms and Internal Stress Balancing

When an external force deforms a solid body, internal restoring forces are generated across the molecular crystal grid. Stress tracks this internal restoring force per unit area, while strain measures the resulting fractional deformation.

Stress and Strain Tensor Types:
Longitudinal Stress / Strain: Acts perpendicular to the cross-section, altering length (Strain = ΔL / L). Can be tensile (stretching) or compressive (squeezing).
 

Shearing Stress / Strain: Acts tangentially across a face, altering shape or angular tilt (Strain = θ ≈ Δx / h).
Volume Stress / Strain: Acts uniformly from all sides, altering total volume (Strain = ΔV / V).

Hooke’s Law: Within the elastic limit, the stress developed in a material is directly proportional to the strain produced:
Stress ∝ Strain ⇒ Stress = E × Strain
(Where E is the Modulus of Elasticity, an inherent property of the material).

Moduli of Elasticity and Stored Energy Density

The specific structural direction of an applied load determines which type of elastic modulus governs the material's response. These configurations also determine how mechanical work is stored within the deformed lattice.

Moduli Proportions and Elastic Energy Integrals

Elastic constants quantify a material's rigidity under various loading scenarios. Integrating stress against strain reveals the potential energy stored within the deformed material.

The Three Elastic Moduli:
Young’s Modulus (Y): Measures resistance to longitudinal stretching:
Y = (Longitudinal Stress) / (Longitudinal Strain) = (F/A) / (ΔL / L) = FL / (AΔL)

Bulk Modulus (B or K): Measures resistance to uniform compression:
B = -(Volume Stress) / (Volume Strain) = -ΔP / (ΔV / V) [Compressibility (β) = 1/B]

Shear Modulus / Rigidity (G or η): Measures resistance to tangentially sliding faces:
η = (Shearing Stress) / (Shearing Strain) = (F_tangential / A) / θ

Poisson’s Ratio (σ): The ratio of lateral strain to longitudinal strain under axial loading. It tracks cross-sectional narrowing during elongation:
σ = (Lateral Strain) / (Longitudinal Strain) = -(Δd / d) / (ΔL / L) [Theoretical Limits: -1 to 0.5; Practical Limits: 0 to 0.5]

Elastic Stored Energy: The work done to deform a solid body is stored as elastic potential energy. This energy can be calculated per unit volume:
U = 1/2 × Stress × Strain = 1/2 × E × (Strain)^2 = 1/2 × (Stress)^2 / E
Total Stored Energy (W) = U × Volume = 1/2 F ΔL

Fluid Statics: Pressure and Hydrostatic Balancing 

A fluid is a substance that cannot sustain a shearing stress when at rest, meaning it deforms continuously under any tangential load. Fluid statics deals with fluids in a state of static equilibrium.

Depth Gradients and Pascal's Transmission
Hydrostatic pressure is isotropic, meaning it exerts an equal force in all directions at any given point. This pressure scales predictably with fluid depth and transfers uniformly across closed liquid systems.

Hydrostatic Pressure Variation with Depth: The pressure in a static liquid increases linearly with depth due to the weight of the fluid column above:
P₂ = P₁ + ρ g h ⇒ P_absolute = P_atmosphere + ρ g h

Gauge Pressure: The difference between absolute pressure and atmospheric pressure (P_gauge = P - P_atm = ρ g h).

Pascal’s Law: Any change in pressure applied to an enclosed, incompressible fluid at rest is transmitted undiminished to every portion of the fluid and to the walls of its container.

Hydraulic Lift Application: A small input force (F₁) applied to a small piston area (A₁) generates a much larger output force (F₂) across a larger piston area (A₂):
F₁ / A₁ = F₂ / A₂ ⇒ F₂ = F₁ (A₂ / A₁)

Pressure in Accelerating Containers (Highly Tested in JEE):

Vertical Acceleration (a_y upward): P = P_atm + ρ(g + a_y)h
Horizontal Acceleration (a_x): The liquid surface tilts at an angle θ relative to the horizontal to remain perpendicular to the effective gravity vector:
tanθ = a_x / g and P(x, y) = P₀ + ρ a_x x + ρ g y

Buoyancy and Archimedes' Principle

When an object is immersed in a fluid, the surrounding pressure increases with depth. This pressure gradient generates a net upward force called the buoyant force.

Displacement Volumes and Flotation Balancing

Archimedes' principle states that the upward buoyant force equals the weight of the fluid displaced by the object. This relationship determines whether an object will sink, float, or remain suspended.

Buoyant Force Vector (F_B): Acts vertically upward through the center of buoyancy (the center of mass of the displaced fluid):
F_B = V_submerged · ρ_fluid · g

Apparent Weight of a Submerged Object:
W_apparent = W_actual - F_B
= V · ρ_body · g - V · ρ_fluid · g
= W_actual(1 - ρ_fluid / ρ_body)

Law of Flotation: For an object floating in static equilibrium at a fluid interface, the weight of the floating body matches the weight of the fluid it displaces:
W_actual = F_B
⇒ V_total · ρ_body · g = V_submerged · ρ_fluid · g
⇒ V_submerged / V_total = ρ_body / ρ_fluid

Fluid Dynamics: Continuity and Bernoulli's Theorem

Fluid dynamics tracks fluids in motion. Ideal fluids are assumed to be steady (non-turbulent), incompressible (constant density), non-viscous (zero internal friction), and irrotational.

Mass Flux Conservation and Energy Density Constants

The behavior of an ideal fluid moving through a pipe is governed by two core principles: the conservation of mass (Continuity) and the conservation of mechanical energy (Bernoulli's Theorem).

Equation of Continuity (Mass Conservation): For an incompressible fluid flowing through a tube of varying cross-sectional area, the volume flow rate (flux) remains constant at all points:
A₁ v₁ = A₂ v₂ ⇒ A · v = Constant
(Where A is the cross-sectional area, and v is the instantaneous fluid velocity. Fluid speeds up where the pipe narrows.

Bernoulli’s Equation (Energy Conservation): For the steady flow of an ideal, incompressible fluid, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along any streamline:
P + 1/2 ρ v² + ρ g h = Constant

Torricelli’s Law of Efflux: The speed at which a liquid flows out of a small hole (orifice) at a depth h below the open surface of an open tank equals the speed a free-falling object would attain falling from the same height:
v = √(2gh)

Time of Flight for the escaping jet to hit the ground:
t = √(2(H - h) / g)
(where H is the total height of the liquid column).

Horizontal Range:
R = v · t = 2√(h(H - h))
Range is maximized when the hole is drilled exactly at the mid-height of the tank (h = H/2).

Viscosity and Stokes' Law

Viscosity measures a fluid's internal resistance to flow, acting as internal friction between adjacent liquid layers that are moving at different speeds.

Velocity Gradients and Drag Equilibria

Viscous forces depend on the relative velocity between fluid layers. When an object falls through a viscous fluid, this drag force increases until it balances gravity, causing the object to reach a constant terminal velocity.

Newton’s Law of Viscosity: The viscous force (F) acting between two parallel fluid layers is directly proportional to their contact area (A) and the velocity gradient (dv/dx) perpendicular to the flow:
F = -η A (dv/dx)
[where η is the Coefficient of Viscosity; SI Unit: Decapoise or Pa·s]

Stokes’ Law: For a tiny solid sphere of radius r moving at speed v through an infinite, uniform fluid with viscosity η, the resistive drag force is:
F_drag = 6π η r v

Terminal Velocity (v_T): The constant, maximum velocity attained by a sphere falling through a viscous fluid, achieved when the net downward gravitational force is perfectly balanced by the upward buoyant and viscous drag forces:
F_viscous + F_buoyancy = F_gravity
⇒ 6π η r v_T + (4/3)π r³ ρ_fluid g = (4/3)π r³ ρ_solid g

v_T = (2/9) · [ r²(ρ_solid - ρ_fluid) g / η ]

Surface Tension and Capillarity

 Surface tension is a molecular property of liquids caused by cohesive forces between molecules. This imbalance of forces draws surface molecules inward, making the liquid surface behave like a stretched elastic membrane.

Free Surface Energies, Excess Pressure, and Meniscus Lifts

Surface tension values track the energy required to expand a liquid's surface area. This surface curvature creates a pressure difference across the boundary, causing liquids to rise or fall inside narrow tubes.

Surface Tension (T or S): Calculated as the force per unit length acting along an imaginary line drawn on the liquid surface, or as the work required to increase the surface area by a unit amount:
T = F / L = dW / dA
[SI Unit: N/m or J/m²]

Excess Pressure Formulations (ΔP = P_concave - P_convex):
Single-Curvature Liquid Meniscus / Curved Surface: ΔP = 2T / R
Liquid Drop in Air (1 Free Surface): ΔP = 2T / R
Air Bubble Inside Liquid (1 Free Surface): ΔP = 2T / R
Soap Bubble in Air (2 Concentric Free Surfaces): ΔP = 4T / R

Capillary Rise (Gurin’s Law): When a narrow glass tube of radius r is dipped into a liquid that wets the glass, the liquid rises to a height h to balance the vertical component of surface tension against the weight of the fluid column:
2π r T cosθ = ρ (π r² h) g
⇒ h = (2T cosθ) / (ρ r g)
(Where θ is the contact angle between the liquid and the tube wall. If θ < 90°, the meniscus is concave and the liquid rises; if θ > 90°, the meniscus is convex and the liquid is depressed.

A clear understanding of the properties of solids and liquids helps in solving a wide range of JEE problems based on elasticity, viscosity, and surface tension. Regular practice of concepts and numerical questions from this chapter can improve both accuracy and confidence in the exam.