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Physics · CDS

PC04 — Gravitation & Properties of Matter

📖 PC04 · CDS General Science — Physics 🎯 CDS Level : High Priority

This chapter spans gravity (from Newton's law to satellites), the properties of solid materials (elasticity, Hooke's law), and fluid mechanics (Pascal's law, Archimedes' principle, surface tension, viscosity). CDS regularly tests Archimedes' principle, buoyancy conditions, and escape velocity.

📌 CDS regularly tests: g variation (height, depth, poles vs equator); escape velocity value (11.2 km/s); orbital velocity value (7.9 km/s); Archimedes' principle (floatation condition); Pascal's law (hydraulic machines); Young's modulus (elasticity); surface tension (capillary rise, insects walking on water); terminal velocity (parachutist).

Topics at a Glance

① Newton's Law of Gravitation

F=GMm/r²; g and its variation

② Orbital & Escape Velocity

7.9 km/s and 11.2 km/s; satellites

③ Elasticity & Hooke's Law

Stress, strain, Young's modulus

④ Fluid Mechanics

Pascal's law; Archimedes; buoyancy

⑤ Surface Tension

Cohesion; capillary rise; contact angle

⑥ Viscosity

Viscous force; terminal velocity; Stokes' law

1. Newton's Law of Gravitation & g

1.1

Gravitational Force & Variation of g

⚡ Gravitation Formulae

Newton's Law: F = G M m / r² G = 6.67 × 10⁻¹¹ N m² kg⁻² (universal — same everywhere; invariant) Surface gravity: g = GM/R² ≈ 9.8 m/s² ≈ 10 m/s² (for CDS calculations) Variation of g: Height h above surface: g' = g(1 − 2h/R) [approx; h << R] Exact: g' = g·R²/(R+h)² Depth d below surface: g' = g(1 − d/R) [linear decrease] At Earth's centre (d=R): g = 0 Latitude (φ): g_pole > g_equator At poles: Earth is oblate (closer to centre) + no centrifugal effect → g highest At equator: farther from centre + centrifugal reduces effective g → g lowest g on Moon ≈ g_Earth / 6 (Moon's mass and radius smaller)

G (universal gravitational constant) never changes. g (local acceleration due to gravity) changes with altitude, depth, and latitude. Weight = mg varies; mass never changes.

2. Orbital Velocity, Escape Velocity & Satellites

2.1

How Objects Orbit & Escape Earth's Gravity

⚡ Satellite Formulae

Orbital velocity (at surface, h=0): v₀ = √(GM/R) ≈ 7.9 km/s Escape velocity: vₑ = √(2GM/R) = √(2gR) ≈ 11.2 km/s Key relation: vₑ = √2 × v₀ ≈ 1.414 × v₀ Kepler's Third Law: T² ∝ r³ (period squared ∝ radius cubed) Period: T = 2π√(r³/GM) Geostationary satellite: Period T = 24 hours (matches Earth's rotation) Altitude ≈ 36,000 km above equator Always appears stationary from Earth → used for TV, weather, communication Weightlessness in orbit: astronauts are in continuous free fall — both astronaut and station fall toward Earth at same rate → no normal force → g_apparent = 0

Escape velocity on Moon ≈ 2.4 km/s (much lower than Earth's 11.2 km/s). This is why the Moon has no atmosphere — gas molecules can escape with thermal velocities.

Fig. 1 — g increases linearly from Earth's centre to the surface; outside the surface it decreases as 1/r². g is maximum at the poles and minimum at the equator.

3. Elasticity — Stress, Strain & Hooke's Law

3.1

How Solid Materials Respond to Forces

⚡ Elasticity Formulae

Stress: σ = F / A [N/m² = Pa] Force per unit area Strain: ε = ΔL / L [dimensionless] Fractional change in length Hooke's Law: Stress ∝ Strain (within elastic limit) Young's Modulus: Y = Stress / Strain = (F/A) / (ΔL/L) = FL / AΔL [Pa] Elastic limit: maximum stress beyond which the material does not return to original shape. Elastic materials: return to original shape (rubber, steel within limit) Plastic materials: permanently deformed (clay, putty)

Steel has a higher Young's modulus than rubber — steel is stiffer (more force needed per unit strain). Rubber stretches easily (low Y) but is elastic. Beyond the elastic limit → material deforms permanently (plastic deformation).

4. Fluid Mechanics — Pascal's Law & Archimedes' Principle

4.1

Pressure in Fluids — Transmission & Buoyancy

Fig. 2 — Left: Pascal's Law — small force on small area creates high pressure, transmitted to large area giving large force (hydraulic press). Right: Archimedes' principle — buoyant force = weight of fluid displaced. If buoyant force ≥ weight → object floats.

⚡ Fluid Mechanics Formulae

Pressure: P = F/A = ρgh [Pascal (Pa)] ρ = fluid density; h = depth below surface Pascal's Law: P₁ = P₂ → F₁/A₁ = F₂/A₂ Applied in: hydraulic press, hydraulic brakes, hydraulic jack Archimedes' Principle: Buoyant force Fb = weight of fluid displaced = ρ_fluid × V_submerged × g Floatation conditions: ρ_object < ρ_fluid → floats (partially submerged) ρ_object = ρ_fluid → neutral buoyancy (submarine at any depth) ρ_object > ρ_fluid → sinks Fraction submerged = ρ_object / ρ_fluid Ice in water: ρ_ice/ρ_water = 0.9/1.0 = 0.9 → 90% submerged, 10% above surface

A ship floats because its average density (including hollow interior with air) is less than water. When a ship is loaded, it sinks deeper but floats as long as average density < water density. Plimsoll line marks the maximum safe loading level.

5. Surface Tension & Viscosity

5.1

Why Droplets Form & Why Fluids Resist Flow

💧 Surface Tension

Force per unit length at the surface of a liquid: T = F/L [N/m]
Caused by: unequal cohesive forces on surface molecules
Cohesion: force between like molecules (water-water)
Adhesion: force between unlike molecules (water-glass)
Surface tension decreases with temperature
Surface tension decreases when soap/detergent is added
Examples: water droplets (spherical shape minimises surface area); insects walking on water; soap bubbles; capillary rise
Capillary rise: liquid rises when adhesion > cohesion (water in glass capillary rises — concave meniscus)
Mercury in glass: cohesion > adhesion → convex meniscus; mercury descends

🐛 Viscosity & Terminal Velocity

Viscosity = internal friction of fluids; resistance to flow
High viscosity: honey, glycerine; Low viscosity: water, air
Viscosity decreases with temperature in liquids (honey flows faster when hot)
Viscosity increases with temperature in gases (opposite of liquids)
Stokes' Law: viscous drag on a sphere: F = 6πηrv
Terminal velocity: when viscous drag + buoyancy = weight → constant velocity downward
Heavier/denser objects → higher terminal velocity
Parachute: large area → large drag → low terminal velocity → safe landing

📝 CDS PYQ

Gravitation & Properties of Matter

Q1. The escape velocity from Earth's surface is approximately:

(a) 7.9 km/s
(b) 9.8 km/s
(c) 11.2 km/s
(d) 3 km/s

Answer: (c) 11.2 km/s
vₑ = √(2gR) = √(2 × 9.8 × 6.4 × 10⁶) ≈ 11.2 km/s. Orbital velocity ≈ 7.9 km/s. Note: vₑ = √2 × v₀ ≈ 1.414 × 7.9 ≈ 11.2 km/s. These two values are among the most tested numerical facts in CDS Physics.

Q2. A block of wood floats with 4/5 of its volume submerged in water. The density of wood is:

(a) 500 kg/m³
(b) 800 kg/m³
(c) 400 kg/m³
(d) 1000 kg/m³

Answer: (b) 800 kg/m³
Fraction submerged = ρ_object / ρ_fluid → 4/5 = ρ_wood / 1000 → ρ_wood = 800 kg/m³. This direct application of the floatation formula is a classic CDS numerical. If density of water = 1000 kg/m³ and 80% is submerged, the wood density must be 800 kg/m³.

Q3. Hydraulic brakes in vehicles work on the principle of:

(a) Archimedes' principle
(b) Pascal's law
(c) Bernoulli's theorem
(d) Newton's third law

Answer: (b) Pascal's Law
Hydraulic brakes work because pressure applied at the brake pedal (small piston, small area) is transmitted equally throughout the fluid and amplified at the wheel cylinders (larger area), applying a much larger force. F₂ = F₁ × A₂/A₁. Pascal's Law: pressure applied to an enclosed fluid is transmitted equally in all directions.

Q4. At what depth inside the Earth does the acceleration due to gravity become zero?

(a) At the surface
(b) At 6400 km depth
(c) At the centre
(d) At 3200 km depth

Answer: (c) At the centre
g decreases linearly as depth d increases inside Earth: g' = g(1 − d/R). At d = R (the centre): g' = g(1 − 1) = 0. At the centre, gravitational pull comes equally from all directions — they cancel out exactly, giving net g = 0. Weight of any body = 0 at Earth's centre, though its mass remains unchanged.

📚 Formula Sheet — PC04

🌎 Gravitation

F = GMm/r²; g = GM/R²
At height h: g' ≈ g(1−2h/R)
At depth d: g' = g(1−d/R)
g=0 at centre; max at poles
v₀ ≈ 7.9 km/s; vₑ ≈ 11.2 km/s

⚡ Elasticity

Stress = F/A [Pa]
Strain = ΔL/L [dimensionless]
Y = Stress/Strain [Pa]
Hooke's law: stress ∝ strain (elastic limit)
Steel Y > rubber Y (stiffer)

💧 Fluid Mechanics

P = ρgh (pressure at depth)
Pascal: F₁/A₁ = F₂/A₂
Archimedes: Fb = ρ_fluid·V·g
Float if ρ_obj < ρ_fluid
Fraction submerged = ρ_obj/ρ_fluid

🐛 Surface Tension & Viscosity

T = F/L [N/m]; decreases with temp
Soap reduces surface tension
Water in glass: rises (adhesion > cohesion)
Mercury: descends (cohesion > adhesion)
Terminal v: viscous drag = weight − buoyancy

⚡ Quick Revision — PC04

🚨 Key Values

v₀ (orbital) ≈ 7.9 km/s
vₑ (escape) ≈ 11.2 km/s
vₑ = √2 × v₀
Geostationary: 24 h; 36,000 km
g_moon = g_earth/6

💧 Floatation

Float: ρ_obj < ρ_fluid
Fraction submerged = ρ_obj/ρ_fluid
Ice: 90% submerged in water
Ship: average density < water
Neutral buoyancy: ρ_obj = ρ_fluid

🚨 CDS Traps

g = 0 at Earth's centre (not deep down)
G never changes; g does
Viscosity of liquids ↓ with temp; gases ↑
Mercury in glass: descends (convex)
Water in glass: rises (concave)

📝 Practice Exercise

E-01

A geostationary satellite has orbital period of:

(a) 6 hours
(b) 12 hours
(c) 24 hours
(d) 90 minutes

E-02

An iron ball sinks in water but floats in mercury. This means:

(a) Density of iron > density of water and mercury
(b) Density of iron > water but < mercury
(c) Density of iron = density of mercury
(d) Density of iron < water

E-03

A liquid rises in a glass capillary tube because:

(a) Cohesive forces > adhesive forces
(b) Adhesive forces > cohesive forces
(c) Gravitational force is zero
(d) Atmospheric pressure pushes it up

E-04

At what height above Earth's surface does g reduce to g/4? (R = radius of Earth)

(a) R/2
(b) R
(c) 2R
(d) 4R

Answers: E-01: (c) 24 hours — geostationary period matches Earth's rotation | E-02: (b) Density of iron > water (sinks) but < mercury (floats) [ρ_water=1000, ρ_iron≈7800, ρ_mercury=13,600 kg/m³] | E-03: (b) Adhesive forces (water-glass) > cohesive forces (water-water) → liquid wets glass and rises | E-04: (b) R [g' = g·R²/(R+h)²; g/4 = g·R²/(R+h)²; (R+h)² = 4R²; R+h = 2R; h = R]

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