Properties of solids and liquids for NEET

"Properties of solids and liquids" is one NMC unit, but it spans three NCERT chapters: mechanical properties of solids, mechanical properties of fluids, and thermal properties of matter. Each part has a handful of formulas, and NEET questions are usually direct applications of one of them, often as a ratio: what happens to the terminal velocity if the radius doubles, or to the capillary rise if the tube is narrower.

That makes the unit a reliable source of marks, provided you know each formula's conditions. This guide takes the three parts in turn, then works through the standard question types.

Elasticity

Stress is restoring force per unit area (N m⁻², same as pressure). Strain is the fractional change in shape or size and has no unit. Hooke's law: within the elastic limit, stress ∝ strain. The constant is a modulus of elasticity.

The reciprocal of bulk modulus is compressibility. From Y = FL/(AΔL), the extension of a wire is ΔL = FL/(AY). For wires of the same material under the same load, ΔL ∝ L/r².

The stress–strain curve in words: stress rises linearly with strain up to the proportional limit (Hooke's law region). Beyond the yield point the material deforms permanently. Stress then reaches a maximum (the ultimate tensile strength) before the wire breaks at the fracture point. Ductile metals have a long gap between the elastic limit and fracture; brittle materials break soon after it. Rubber stretches a great deal without a linear region.

Fluids at rest

• Pressure at depth h: P = P₀ + ρgh. The gauge pressure is ρgh. Pressure depends on depth, not on the shape of the container.
• Pascal's law: a change in pressure applied to an enclosed fluid is transmitted undiminished to every point. In a hydraulic lift, F₁/A₁ = F₂/A₂, so a small force on a small piston lifts a large load on a large one.
• Archimedes' principle: the upthrust on a body equals the weight of fluid it displaces.

Fluids in motion

• Continuity (incompressible, steady flow): A₁v₁ = A₂v₂. Where a pipe narrows, the fluid speeds up.
• Bernoulli's principle (steady, non-viscous, incompressible flow along a streamline): P + ½ρv² + ρgh = constant. Where speed rises, pressure falls.
• Speed of efflux (Torricelli) from a hole at depth h below the free surface: v = √(2gh).

Bernoulli explains the lift on an aircraft wing and why a strong wind can lift a roof. Flow is streamline at low speeds and turns turbulent above a critical velocity. Older textbooks go further, with the Venturi meter and the Reynolds number (Re = ρvd/η, roughly below 1,000 streamline and above 2,000 turbulent); the rationalised NCERT has dropped both, so treat them as background rather than core syllabus.

Viscosity and terminal velocity

Viscosity is fluid friction between layers. The viscous force is F = ηA(dv/dx), where η is the coefficient of viscosity (SI unit Pa s; 1 Pa s = 10 poise).

• Stokes' law: a small sphere moving slowly through a viscous fluid feels F = 6πηrv.
• Terminal velocity: when weight = upthrust + viscous drag, v_t = 2r²(ρ − σ)g / 9η, where ρ is the sphere's density and σ the fluid's. So v_t ∝ r².

Surface tension

Surface tension T is force per unit length along a line on the surface, and equally the surface energy per unit area (unit N m⁻¹ or J m⁻²).

• Angle of contact θ: acute when the liquid wets the solid (water on clean glass), obtuse when it does not (mercury on glass).
• Excess pressure inside: a liquid drop 2T/r; an air bubble inside a liquid 2T/r; a soap bubble 4T/r, because it has two surfaces.
• Capillary rise: h = 2T cos θ / (rρg). The liquid rises if θ < 90° and is depressed if θ > 90°. Narrower tubes give higher rise (h ∝ 1/r).
• Work to blow a soap bubble of radius r = T × 2 × 4πr² = 8πr²T (two surfaces again).

Thermal properties of matter

Calorimetry rests on one idea: in an insulated mixture, heat lost by the hot bodies equals heat gained by the cold ones. Convection is heat carried by the movement of the fluid itself, as in sea breezes.

Worked numericals

Take g = 10 m s⁻² and ρ_water = 1,000 kg m⁻³.

Example 1: Young's modulus of a wire

A wire 2 m long with cross-section 1 mm² stretches by 1 mm under 100 N.

• Stress = 100 / 10⁻⁶ = 10⁸ N m⁻². Strain = 10⁻³ / 2 = 5 × 10⁻⁴.
• Y = 10⁸ / (5 × 10⁻⁴) = 2 × 10¹¹ N m⁻², a typical value for steel.

Example 2: continuity and Bernoulli

Water flows at 2 m s⁻¹ through a horizontal pipe, which narrows to one-third of the area.

• Continuity: v₂ = 2 × 3 = 6 m s⁻¹.
• Bernoulli (same height): P₁ − P₂ = ½ρ(v₂² − v₁²) = ½ × 1,000 × (36 − 4) = 16,000 Pa.

Example 3: drops merging

Eight identical raindrops, each falling at terminal velocity v, merge into one drop. Find its terminal velocity.

• Volume is conserved: (4/3)πR³ = 8 × (4/3)πr³, so R = 2r.
• v_t ∝ r², so the new terminal velocity is 4v.

Example 4: ice in warm water

50 g of ice at 0 °C is dropped into 200 g of water at 40 °C. Find the final temperature. (Take L = 80 cal g⁻¹ and c = 1 cal g⁻¹ °C⁻¹; ignore the container.)

• Heat the warm water gives by cooling to 0 °C = 200 × 1 × 40 = 8,000 cal.
• Heat needed to melt all the ice = 50 × 80 = 4,000 cal. So all the ice melts, leaving 4,000 cal.
• This warms all 250 g from 0 °C: 4,000 / 250 = 16 °C.
• Check: water cooling 40 → 16 loses 200 × 24 = 4,800 cal; ice gains 4,000 + 50 × 16 = 4,800 cal ✓.

Practice MCQs

1. A drop of radius r falls at terminal velocity v. A drop of radius 2r of the same liquid falls at: (a) 8v (b) 4v (c) 2v (d) v
2. The gauge pressure 10 m below the surface of water is: (a) 10³ Pa (b) 10⁴ Pa (c) 10⁵ Pa (d) 10⁶ Pa
3. A hydraulic lift has pistons of area 0.01 m² and 0.5 m². The force needed on the small piston to lift a 15,000 N car is: (a) 30 N (b) 150 N (c) 750 N (d) 300 N
4. Two wires of the same material have lengths in the ratio 1 : 2 and diameters in the ratio 1 : 2. Under the same load, their extensions are in the ratio: (a) 2 : 1 (b) 1 : 2 (c) 1 : 1 (d) 1 : 4
5. The excess pressure inside a soap bubble of radius 1 cm (T = 0.03 N m⁻¹) is: (a) 12 Pa (b) 6 Pa (c) 3 Pa (d) 24 Pa
6. If the radius of a capillary tube is halved, the height of water rise becomes: (a) half (b) the same (c) double (d) four times
7. Water leaks from a small hole 5 m below the surface of a tank. The speed of efflux is: (a) 5 m s⁻¹ (b) 7 m s⁻¹ (c) 10 m s⁻¹ (d) 50 m s⁻¹
8. The absolute temperature of a black body is doubled. The power it radiates becomes: (a) 2 times (b) 4 times (c) 8 times (d) 16 times

Answers

1. (b) v_t ∝ r², so 2² = 4 times.
2. (c) ρgh = 1,000 × 10 × 10 = 10⁵ Pa.
3. (d) F = 15,000 × 0.01 / 0.5 = 300 N.
4. (a) ΔL ∝ L/d²: (1/1) : (2/4) = 1 : 0.5 = 2 : 1.
5. (a) 4T/r = 4 × 0.03 / 0.01 = 12 Pa.
6. (c) h ∝ 1/r.
7. (c) v = √(2 × 10 × 5) = 10 m s⁻¹.
8. (d) H ∝ T⁴, and 2⁴ = 16.

What to do next

• Make one formula page for this unit in three blocks (solids, fluids, thermal), with the condition for each formula written beside it.
• Practise ratio questions: terminal velocity, capillary rise, extension of wires, radiated power.
• Solve five calorimetry problems with a change of state, and check each with the heat-lost = heat-gained balance.
• Carry the thermal ideas forward to thermodynamics for NEET.

For the gravity behind fluid pressure, see gravitation for NEET.