📌 1.1 Basic Rotational Quantities
Angular Displacement (θ)
Angle swept by radius vector. Unit: radian.
θ = arc / radius = s/r
Angular Velocity (ω)
Rate of change of angular displacement.
ω = dθ/dt = v/r (rad/s)
Angular Acceleration (α)
Rate of change of angular velocity.
α = dω/dt = a/r (rad/s²)
Period & Frequency
Time for one revolution.
T = 2π/ω n = 1/T = ω/2π
Analogies: Linear ↔ Rotational
s ↔ θ | v ↔ ω | a ↔ α | m ↔ I | F ↔ τ | p ↔ L | KE = ½mv² ↔ ½Iω²
📐 1.2 Equations of Rotational Motion
For uniform angular acceleration (α = constant):
2nd Equation
θ = ω₀t + ½αt²
3rd Equation
ω² = ω₀² + 2αθ
Average ω
ω_av = (ω₀ + ω) / 2
⭐ These are exactly analogous to v = u+at, s = ut+½at², v² = u²+2as. Replace v→ω, u→ω₀, a→α, s→θ.
⚡ 1.3 Torque (Moment of Force)
Torque Formula
τ⃗ = r⃗ × F⃗ |τ| = r·F·sinθ Unit: N·m
- Torque is the rotational equivalent of force
- τ = r × F × sinθ where θ is angle between r⃗ and F⃗
- Torque is maximum when F ⊥ r (θ = 90°) → τ = rF
- Torque is zero when F ∥ r (θ = 0°)
- τ = I × α (analogous to F = ma)
Newton's 2nd Law — Rotational Form
τ_net = I · α (analogous to F_net = m·a)
⭐ Couple: Two equal and opposite parallel forces acting on a body constitute a couple. Torque of couple = F × d (d = perpendicular distance between forces). Net force of couple = 0, but torque ≠ 0.
🔩 1.4 Moment of Inertia (I)
Definition
I = Σmᵢrᵢ² = ∫r² dm Unit: kg·m²
I is the rotational equivalent of mass. It depends on mass, shape and axis of rotation.
| Body | Axis | Moment of Inertia I |
|---|
| Thin rod (length L) | Through centre ⊥ length | I = ML²/12 |
| Thin rod (length L) | Through one end ⊥ length | I = ML²/3 |
| Solid cylinder / disc (radius R) | Through centre (axis) | I = MR²/2 |
| Hollow cylinder (radius R) | Through centre (axis) | I = MR² |
| Solid sphere (radius R) | Through diameter | I = 2MR²/5 |
| Hollow sphere (radius R) | Through diameter | I = 2MR²/3 |
| Thin ring / hoop (radius R) | Through centre ⊥ plane | I = MR² |
| Thin ring (radius R) | Through diameter | I = MR²/2 |
| Rectangular plate (a×b) | Through centre | I = M(a²+b²)/12 |
📏 1.5 Theorems on Moment of Inertia
Parallel Axis Theorem
I about any axis = I_cm + Md²
where d = distance between the two parallel axes. One axis must pass through CM.
I = I_cm + Md²
Perpendicular Axis Theorem
For flat lamina only: I_z = I_x + I_y
z-axis ⊥ to plane of lamina. x and y in the plane of lamina.
I_z = I_x + I_y
⭐ Example: I of disc about diameter = MR²/4 (using perp. axis: I_z = MR²/2 = I_x + I_y = 2I_diam → I_diam = MR²/4)
🌀 1.6 Angular Momentum (L)
Angular Momentum
L⃗ = r⃗ × p⃗ = I·ω⃗ |L| = Iω = mvr·sinθ Unit: kg·m²/s
Newton's 2nd Law
Torque = rate of change of angular momentum
τ = dL/dt
Conservation of L
If τ_net = 0, then L = constant
I₁ω₁ = I₂ω₂
⭐ Conservation of Angular Momentum Examples:
1. Ice skater pulls arms in → I decreases → ω increases (spin faster)
2. Diver tucks → I decreases → ω increases → more rotations
3. Earth's revolution: at perihelion (closer to Sun) it moves faster — L conserved
🎱 1.7 Rolling Motion (Without Slipping)
Condition for Rolling Without Slipping
v_cm = R·ω a_cm = R·α
Total KE (Rolling)
KE = ½Mv² + ½Iω²
= ½Mv²(1 + I/MR²)
For Solid Sphere
KE = ½Mv²(1 + 2/5) = 7/10 Mv²
For Disc/Cylinder
KE = ½Mv²(1 + 1/2) = 3/4 Mv²
For Ring/Hoop
KE = ½Mv²(1 + 1) = Mv²
| Body | I/MR² | v at bottom of incline | Reaches bottom |
|---|
| Solid sphere | 2/5 | v = √(10gh/7) | First (fastest) |
| Solid cylinder | 1/2 | v = √(4gh/3) | Second |
| Hollow sphere | 2/3 | v = √(6gh/5) | Third |
| Ring / Hoop | 1 | v = √(gh) | Last (slowest) |
⭐ Key principle: Body with smaller I/MR² reaches the bottom first. Solid sphere always wins in a rolling race.
⚖️ 1.8 Conditions for Equilibrium
Translational Equilibrium
Vector sum of all forces = 0
ΣF⃗ = 0 (ΣFx=0, ΣFy=0)
Rotational Equilibrium
Vector sum of all torques = 0
Στ = 0
⭐ Principle of Moments: For a body in equilibrium, sum of clockwise moments = sum of anticlockwise moments about any point.
⚡ 1.9 Work, Power & Energy — Rotation
Work done by Torque
W = τ·θ (for const. torque)
W = ∫τ dθ (variable)
Rotational KE
KE_rot = ½Iω²
Work-Energy Theorem
W_net = ΔKE_rot = ½I(ω²−ω₀²)
🌡️ 2.1 Basic Thermodynamic Concepts
System & Surroundings
System = object under study. Surroundings = everything else. Boundary separates them.
Thermodynamic State
Described by state variables: P, V, T, n. State functions depend only on current state, not on path.
Internal Energy (U)
Total KE + PE of all molecules. State function. For ideal gas: depends only on T.
Thermal Equilibrium
No net heat flow between two bodies in contact. Zeroth Law basis.
Sign Convention
Q > 0 → heat absorbed by system | Q < 0 → heat released by system
W > 0 → work done BY system | W < 0 → work done ON system
0️⃣ 2.2 Zeroth Law of Thermodynamics
Zeroth Law
If body A is in thermal equilibrium with body B, and B is in equilibrium with body C,
then A and C are also in thermal equilibrium with each other.
This law defines the concept of temperature. It is the basis for all thermometers — a thermometer measures temperature by reaching thermal equilibrium with the body.
1️⃣ 2.3 First Law of Thermodynamics
🌟 First Law (Conservation of Energy)
ΔU = Q − W OR Q = ΔU + W
dU = dQ − dW dW = P dV
Work done BY gas during expansion:
W = ∫P dV = Area under P-V diagram
| Process | Condition | W | ΔU | Q |
|---|
| Isothermal | T = const (ΔT=0) | nRT ln(V₂/V₁) | 0 | Q = W |
| Isochoric | V = const (ΔV=0) | 0 | nCvΔT | Q = ΔU = nCvΔT |
| Isobaric | P = const (ΔP=0) | PΔV = nRΔT | nCvΔT | nCpΔT |
| Adiabatic | Q = 0 (no heat) | −ΔU = nCvΔT | −W = nCvΔT | 0 |
⭐ Adiabatic process equations: PV^γ = const | TV^(γ-1) = const | T^γ P^(1-γ) = const
🔥 2.4 Specific Heat Capacities of Gases
Cv (constant volume)
Heat needed per mole per degree K at constant volume.
Cv = ΔU/nΔT
Cp (constant pressure)
Heat needed per mole per degree K at constant pressure.
Cp = ΔQ/nΔT
Mayer's Relation
Cp is always greater than Cv by R.
Cp − Cv = R
Ratio γ
Adiabatic index (Heat capacity ratio)
γ = Cp/Cv
Mono: 5/3, Di: 7/5
| Type of Gas | Degrees of Freedom (f) | Cv | Cp | γ |
|---|
| Monatomic (He, Ar) | 3 | 3R/2 | 5R/2 | 5/3 ≈ 1.67 |
| Diatomic (N₂, O₂, H₂) | 5 | 5R/2 | 7R/2 | 7/5 = 1.4 |
| Polyatomic (CO₂) | 6 | 3R | 4R | 4/3 ≈ 1.33 |
⭐ Equipartition of Energy: Each degree of freedom contributes ½kT to the average energy. Total internal energy U = f/2 · nRT
2️⃣ 2.5 Second Law of Thermodynamics
Kelvin-Planck Statement
It is impossible to construct a heat engine that absorbs heat from a single reservoir and converts it completely into work with no other effect.
Clausius Statement
Heat cannot spontaneously flow from a colder body to a hotter body without external work being done on the system.
Entropy (S)
dS = dQ_rev / T ΔS ≥ 0 (for isolated system)
Entropy of universe always increases for irreversible processes.
⭐ The Second Law means: Perfect heat engine (100% efficiency) is impossible. Natural processes are irreversible. Entropy defines the direction of natural processes.
⚙️ 2.6 Heat Engines & Efficiency
Heat Engine
Q₁ = heat absorbed (from hot reservoir) | Q₂ = heat rejected (to cold reservoir)
W = Q₁ − Q₂ | η = W/Q₁ = 1 − Q₂/Q₁ = 1 − T₂/T₁ (Carnot)
Carnot Engine
Most efficient engine between two temperatures. Consists of 2 isothermal + 2 adiabatic processes.
η_Carnot = 1 − T₂/T₁
Refrigerator (COP)
Coefficient of Performance = heat removed / work input
COP = Q₂/W = T₂/(T₁−T₂)
Heat Pump (COP)
COP of heat pump = heat delivered to hot reservoir / work input
COP_hp = Q₁/W = T₁/(T₁−T₂)
Carnot Theorem
No engine operating between two reservoirs can be more efficient than the Carnot engine.
η_any ≤ η_Carnot
⭐ Carnot Cycle (4 steps):
1. Isothermal expansion at T₁ (absorbs Q₁) → 2. Adiabatic expansion (T₁→T₂) → 3. Isothermal compression at T₂ (rejects Q₂) → 4. Adiabatic compression (T₂→T₁)
📊 2.7 PV Diagrams — Quick Reference
| Process | PV Diagram Shape | Slope |
|---|
| Isothermal | Rectangular hyperbola (PV = const) | Slope = −P/V |
| Adiabatic | Steeper hyperbola (PV^γ = const) | Slope = −γP/V |
| Isochoric | Vertical straight line (V = const) | Infinite (vertical) |
| Isobaric | Horizontal straight line (P = const) | Zero (horizontal) |
⭐ Adiabatic curve is always steeper than isothermal curve at the same point. Area under PV curve = Work done by/on gas.
🔩 3.1 Elasticity — Stress & Strain
Stress
Restoring force per unit area. Unit: N/m² = Pascal (Pa)
Stress = F/A
Strain
Fractional change in dimension. Dimensionless.
Strain = ΔL/L (or ΔV/V or Δx/L)
Hooke's Law
Within elastic limit, stress ∝ strain
Stress / Strain = Modulus (constant)
Elastic Limit
Maximum stress beyond which material does not return to original shape.
| Type of Stress | Type of Strain | Modulus | Formula |
|---|
| Longitudinal (Tensile) | Longitudinal (ΔL/L) | Young's Modulus (Y) | Y = (F/A)/(ΔL/L) = FLₒ/AΔL |
| Tangential (Shear) | Shear (Δx/L = tanφ) | Modulus of Rigidity (η or G) | η = (F/A)/(Δx/L) |
| Volume (Hydraulic) | Volumetric (ΔV/V) | Bulk Modulus (K or B) | K = −P/(ΔV/V) |
Compressibility
Compressibility = 1/K = −(1/V)(dV/dP) Unit: Pa⁻¹
⭐ Poisson's Ratio (σ): σ = −(lateral strain) / (longitudinal strain) = −(ΔD/D) / (ΔL/L). Range: −1 ≤ σ ≤ 0.5. For most materials σ ≈ 0.25–0.35.
📈 3.2 Stress-Strain Curve
- O to A (Proportional Limit): Stress ∝ Strain — Hooke's Law valid, perfectly elastic
- A to B (Elastic Limit): No longer proportional, but still elastic (returns to original)
- B (Yield Point): Permanent deformation begins — material becomes plastic
- B to D (Plastic Region): Large deformation for small stress increase
- D (Ultimate Tensile Strength): Maximum stress the material can withstand
- D to E (Fracture Point): Material necks down and breaks
Ductile Materials
Large plastic region before fracture. Can be drawn into wires. e.g. copper, steel, gold
Brittle Materials
Very small plastic region — break suddenly. e.g. glass, cast iron, ceramics
Elastomers
Very large elastic range, low Young's modulus. e.g. rubber
⭐ Elastic Potential Energy: U = ½ × stress × strain × volume = ½ × (F/A) × (ΔL/L) × AL = F·ΔL/2. Energy density = ½ × Y × (strain)²
💧 3.3 Mechanical Properties of Fluids — Pressure
Fluid Pressure
P = F/A P = P₀ + ρgh (gauge pressure = ρgh)
Pascal's Law
Pressure applied to an enclosed fluid is transmitted equally in all directions.
P₁ = P₂ (hydraulic press)
Archimedes' Principle
Buoyant force = weight of fluid displaced.
F_b = ρ_fluid × V_submerged × g
Condition for Floating
Object floats when buoyant force = weight.
ρ_object ≤ ρ_fluid
Atmospheric Pressure
Standard: 1 atm = 101325 Pa = 760 mm Hg
1 atm = 1.013 × 10⁵ Pa
🌊 3.4 Fluid Dynamics — Bernoulli's Theorem
Equation of Continuity
A₁v₁ = A₂v₂ (for incompressible fluid)
🌟 Bernoulli's Equation
P + ½ρv² + ρgh = constant (along a streamline)
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Applications of Bernoulli's Theorem:
- Venturimeter: Measures flow speed using pressure difference at narrowing. v = A₂√[2(P₁-P₂)/ρ(A₁²-A₂²)]
- Torricelli's Theorem: Speed of efflux from a hole = √(2gh). v = √(2gh)
- Aerofoil/Wing lift: Air moves faster over curved top → less pressure above → lift
- Atomiser/Sprayer: Fast air stream above liquid → reduced pressure → liquid rises
- Magnus Effect: Spinning ball curves in flight due to pressure difference
⭐ Bernoulli's equation is essentially the energy conservation equation for flowing fluids. It applies to non-viscous, incompressible, irrotational, steady flow.
🫧 3.5 Viscosity
Newton's Law of Viscosity
F = η · A · (dv/dy) η = viscosity coefficient, Unit: Pa·s (or Poise)
Stokes' Law
Drag on sphere moving through viscous fluid
F = 6πηrv
Terminal Velocity
When drag + buoyancy = weight. Sphere falls at constant speed.
v_t = 2r²(ρ-ρ₀)g / 9η
Reynolds Number (Re)
Ratio of inertial to viscous forces. Predicts flow type.
Re = ρvD/η
Flow Types
Re < 1000 → Laminar. Re > 2000 → Turbulent. 1000–2000 → Transitional.
Poiseuille's Formula — Flow through pipe
Q = πPr⁴ / 8ηl (volume flow rate, P = pressure difference)
⭐ Viscosity of liquids DECREASES with temperature. Viscosity of gases INCREASES with temperature.
🫧 3.6 Surface Tension
Surface Tension (T or S)
T = F/l = Surface energy per unit area Unit: N/m or J/m²
Excess Pressure in Bubble
Soap bubble has 2 surfaces (inner + outer).
ΔP_soap = 4T/R
ΔP_liquid drop = 2T/R
Capillary Rise
Height liquid rises in capillary tube.
h = 2T cosθ / (ρgr)
Angle of Contact (θ)
θ < 90° → liquid wets surface (water-glass)
θ > 90° → liquid doesn't wet (mercury-glass)
Surface Energy
Work done in increasing surface area.
W = T × ΔA
⭐ Capillary action: Water rises in plant stems, ink spreads in paper, water absorbed by towel — all due to surface tension and capillarity.
⭐ Effect of temperature: Surface tension DECREASES with increase in temperature. At critical temperature, surface tension = 0.
📋 3.7 Mechanical Properties — Complete Formula Sheet
| Quantity | Formula | Unit |
|---|
| Young's Modulus Y | Y = (F × L₀) / (A × ΔL) | N/m² (Pa) |
| Bulk Modulus K | K = −P / (ΔV/V) | N/m² (Pa) |
| Modulus of Rigidity η | η = (F/A) / (Δx/L) | N/m² (Pa) |
| Pressure P | P = P₀ + ρgh | Pa |
| Buoyant Force | F_b = ρ_f V_s g | N |
| Bernoulli | P + ½ρv² + ρgh = const | — |
| Continuity | A₁v₁ = A₂v₂ | — |
| Terminal velocity | v_t = 2r²(ρ-ρ₀)g / 9η | m/s |
| Capillary rise | h = 2T cosθ / (ρgr) | m |
| Excess pressure (soap) | ΔP = 4T/R | Pa |
| Excess pressure (drop) | ΔP = 2T/R | Pa |
| Stokes drag | F = 6πηrv | N |
| Efflux speed | v = √(2gh) | m/s |
🔄 Rotational — Must Know
- All MI formulae from the table
- Parallel + Perpendicular axis theorems
- Rolling race — solid sphere wins
- L = Iω and its conservation
- τ = Iα = dL/dt
- Work done = τθ, Power = τω
🌡️ Thermodynamics — Must Know
- All 4 process formulas (W, ΔU, Q)
- Carnot efficiency = 1 − T₂/T₁
- Cp − Cv = R (Mayer's relation)
- γ values for mono, di, polyatomic
- Adiabatic: PV^γ = const
- Entropy: ΔS ≥ 0
⚙️ Mechanical Props — Must Know
- Y, K, η definitions and units
- Bernoulli's equation application
- Excess pressure: 4T/R vs 2T/R
- Terminal velocity formula
- Capillary rise formula
- Viscosity ↓ with temp (liquids)
⚠️ Common Mistakes:
1. Forgetting factor of 2 in soap bubble (2 surfaces → 4T/R not 2T/R)
2. Using wrong MI formula — axis matters! Rod through centre vs end gives different I
3. In adiabatic: Q = 0, NOT ΔU = 0. Isothermal: ΔU = 0, NOT Q = 0
4. Carnot efficiency formula needs absolute temperature (Kelvin), not Celsius
5. In rolling: Total KE = Translational KE + Rotational KE — don't forget rotational part
6. Angular momentum L = Iω, NOT L = mvr (that's for a particle, use L = mvr sinθ)