Mechanics Mindmap - Comprehensive Visual Guide

📋 Introduction

This mechanics mindmap provides a visual overview of all classical mechanics concepts, formulas, and problem-solving techniques essential for JEE Advanced preparation. It covers kinematics, dynamics, work-energy, rotational motion, and gravitation.

🎯 Mechanics Framework

Main Branches:

Mechanics
├── Kinematics
├── Newton's Laws of Motion
├── Work, Energy and Power
├── Rotational Motion
├── Gravitation
└── Properties of Matter

🚀 Kinematics

Kinematics Overview:

Kinematics
├── Motion in One Dimension
│   ├── Position, Displacement, Distance
│   ├── Velocity and Speed
│   ├── Acceleration
│   ├── Equations of Motion
│   ├── Graphical Analysis
│   └── Relative Motion
├── Motion in Two Dimensions
│   ├── Vector Representation
│   ├── Projectile Motion
│   ├── Circular Motion
│   └── Relative Velocity
├── Motion in Three Dimensions
│   ├── Vector Analysis
│   ├── Curvilinear Motion
│   └── Angular Motion
└── Graphical Kinematics
    ├── Position-Time Graphs
    ├── Velocity-Time Graphs
    ├── Acceleration-Time Graphs
    └── Area Under Curves

Kinematics Key Formulas:

Essential Kinematics Formulas:
1. Equations of Motion (Uniform Acceleration):
   - v = u + at
   - s = ut + ½at²
   - v² = u² + 2as
   - s = vt - ½at²
   - s = [(u + v)/2]t

2. Projectile Motion:
   - Time of flight: T = 2u sin θ/g
   - Maximum height: H = u² sin²θ/2g
   - Range: R = u² sin 2θ/g
   - At any time t: x = u cos θ·t, y = u sin θ·t - ½gt²

3. Circular Motion:
   - Angular velocity: ω = v/r
   - Centripetal acceleration: a = v²/r = ω²r
   - Period: T = 2πr/v = 2π/ω
   - Frequency: f = 1/T = ω/2π

4. Relative Motion:
   - Relative velocity: v_AB = v_A - v_B
   - River boat problems: Effective velocity
   - Rain drop problems: Relative angles

⚖️ Newton’s Laws of Motion

Newton’s Laws Overview:

Newton's Laws of Motion
├── First Law (Inertia)
│   ├── Concept of Inertia
│   ├── Frame of Reference
│   ├── Inertial and Non-inertial Frames
│   └── Applications
├── Second Law (F = ma)
│   ├── Force and Acceleration
│   ├── Momentum
│   ├── Impulse
│   └── Force-Time Graphs
├── Third Law (Action-Reaction)
│   ├── Action-Reaction Pairs
│   ├── Examples and Applications
│   └── Common Misconceptions
├── Free Body Diagrams
│   ├── Drawing FBDs
│   ├── Identifying Forces
│   ├── Force Resolution
│   └── Equilibrium Conditions
├── Types of Forces
│   ├── Contact Forces
│   │   ├── Normal Force
│   │   ├── Friction
│   │   ├── Tension
│   │   └── Spring Force
│   ├── Non-contact Forces
│   │   ├── Gravitational Force
│   │   ├── Electric Force
│   │   └── Magnetic Force
│   └── Pseudo Forces
│       ├── Centrifugal Force
│       └── Coriolis Force
└── Applications of Newton's Laws
    ├── Connected Bodies
    ├── Pulley Systems
    ├── Inclined Planes
    ├── Circular Motion Dynamics
    └── Banking of Roads

Force Analysis Framework:

Systematic Force Analysis:
1. Draw Free Body Diagram:
   - Identify the object
   - Draw all forces
   - Use proper direction
   - Label forces clearly

2. Apply Newton's Second Law:
   - ΣF = ma (in x-direction)
   - ΣF = ma (in y-direction)
   - Resolve forces if needed
   - Consider sign conventions

3. Friction Forces:
   - Static friction: f_s ≤ μ_s N
   - Kinetic friction: f_k = μ_k N
   - Direction: Opposes relative motion
   - Maximum static friction: f_max = μ_s N

4. Pulley Systems:
   - Same tension throughout string
   - Consider mass of pulley if significant
   - Account for friction at pulley
   - Multiple pulleys: tension distribution

⚡ Work, Energy and Power

Work-Energy Overview:

Work, Energy and Power
├── Work
│   Definition and Mathematical Expression
│   Types of Work
│   │   ├── Positive Work
│   │   ├── Negative Work
│   │   └── Zero Work
│   Work Done by Variable Force
│   Work-Energy Theorem
│   └── Conservative and Non-conservative Forces
├── Energy
│   ├── Kinetic Energy
│   │   ├── Translational KE
│   │   └── Rotational KE
│   ├── Potential Energy
│   │   ├── Gravitational PE
│   │   ├── Elastic PE
│   │   └── Electric PE
│   ├── Mechanical Energy
│   ├── Conservation of Energy
│   └── Energy Transformations
├── Power
│   Definition and Units
│   Average and Instantaneous Power
│   Power in Circular Motion
│   └── Efficiency
├── Collisions
│   ├── Types of Collisions
│   │   ├── Elastic Collisions
│   │   ├── Inelastic Collisions
│   │   └── Perfectly Inelastic Collisions
│   ├── Conservation Laws in Collisions
│   ├── Coefficient of Restitution
│   └── Applications
└── Applications of Work-Energy
    ├── Simple Machines
    ├── Projectiles with Air Resistance
    ├── Energy Diagrams
    └── Stability and Potential Energy

Energy Conservation Principles:

Energy Conservation Framework:
1. Work Done by Force:
   - Constant force: W = F·d·cosθ
   - Variable force: W = ∫F·ds
   - Work by gravity: W = mgh
   - Work by spring: W = ½k(x₂² - x₁²)

2. Kinetic Energy:
   - Translational: KE = ½mv²
   - Rotational: KE = ½Iω²
   - Total: KE = ½mv² + ½Iω²

3. Potential Energy:
   - Gravitational: PE = mgh
   - Spring: PE = ½kx²
   - Electric: PE = kq₁q₂/r

4. Conservation of Energy:
   - E_total = KE + PE = constant
   - ΔKE + ΔPE = 0 (conservative forces only)
   - W_nc = ΔKE + ΔPE (with non-conservative forces)

5. Power Calculations:
   - Instantaneous: P = dW/dt = F·v
   - Average: P = W/t
   - Rotational: P = τ·ω

🔄 Rotational Motion

Rotational Motion Overview:

Rotational Motion
├── Rotational Kinematics
│   ├── Angular Displacement, Velocity, Acceleration
│   ├── Equations of Rotational Motion
│   ├── Relation Between Linear and Angular Quantities
│   └── Angular Kinematics Graphs
├── Rotational Dynamics
│   ├── Moment of Inertia
│   │   ├── Definition
│   │   ├── Calculation Methods
│   │   ├── Parallel Axis Theorem
│   │   └── Perpendicular Axis Theorem
│   ├── Torque
│   │   ├── Definition and Direction
│   │   ├── Torque Due to Forces
│   │   └── Torque and Angular Acceleration
│   ├── Angular Momentum
│   │   ├── Definition and Conservation
│   │   ├── Angular Momentum of Rigid Bodies
│   │   └── Relation with Linear Momentum
│   └── Rotational Dynamics Equations
├── Rolling Motion
│   ├── Pure Rolling
│   ├── Rolling Without Slipping
│   ├── Rolling with Slipping
│   └── Energy in Rolling Motion
├── Angular Momentum Conservation
│   ├── Conservation Principles
│   ├── Applications
│   ├── Gyroscope
│   └── Precession
└── Applications of Rotational Motion
    ├── Figure Skater Effect
    ├── Diving and Gymnastics
    ├── Rotational Equilibrium
    └── Rotating Reference Frames

Rotational Motion Key Concepts:

Essential Rotational Formulas:
1. Angular Kinematics:
   - ω = ω₀ + αt
   - θ = ω₀t + ½αt²
   - ω² = ω₀² + 2αθ
   - v = rω (linear-angular relationship)
   - a = rα (linear-angular relationship)

2. Moment of Inertia:
   - I = Σmr² (point masses)
   - I = ∫r²dm (continuous distribution)
   - Parallel axis: I = I_cm + Md²
   - Perpendicular axis: I_z = I_x + I_y (lamina)

3. Common Moments of Inertia:
   - Rod (about center): I = ML²/12
   - Rod (about end): I = ML²/3
   - Ring: I = MR²
   - Disk: I = MR²/2
   - Sphere: I = 2MR²/5

4. Torque and Angular Momentum:
   - τ = r × F = Iα
   - L = r × p = Iω
   - τ = dL/dt
   - Conservation: L = constant (if τ = 0)

5. Rolling Motion:
   - v_cm = ωR (pure rolling)
   - KE_total = ½mv² + ½Iω²
   - KE_trans = ½mv²
   - KE_rot = ½Iω²

🌍 Gravitation

Gravitation Overview:

Gravitation
├── Universal Law of Gravitation
│   ├── Newton's Law of Gravitation
│   ├── Gravitational Constant G
│   ├── Vector Form of Gravitational Force
│   └── Superposition Principle
├── Gravitational Field
│   ├── Gravitational Field Strength
│   ├── Field Due to Various Mass Distributions
│   ├── Gravitational Potential
│   └── Equipotential Surfaces
├── Gravitational Potential Energy
│   ├── Binding Energy
│   ├── Escape Velocity
│   └── Orbital Energy
├── Motion of Planets and Satellites
│   ├── Kepler's Laws
│   │   ├── First Law (Law of Ellipses)
│   │   ├── Second Law (Law of Areas)
│   │   └── Third Law (Law of Periods)
│   ├── Orbital Velocity
│   ├── Orbital Period
│   └── Satellite Motion
├── Earth's Gravitational Field
│   ├── Earth's Shape and Gravity
│   ├── Variation of g with Height
│   ├── Variation of g with Depth
│   └── Effect of Earth's Rotation
└── Applications of Gravitation
    ├── Geostationary Satellites
    ├── GPS Systems
    ├── Planetary Motion
    └── Tidal Forces

Gravitation Key Formulas:

Essential Gravitation Formulas:
1. Newton's Law of Gravitation:
   - F = G(m₁m₂/r²)
   - G = 6.67 × 10^-11 N·m²/kg²
   - Vector form: F = -G(m₁m₂/r²)r̂

2. Gravitational Field:
   - Field strength: g = F/m = GM/r²
   - Gravitational potential: V = -GM/r
   - Field and potential: g = -dV/dr

3. Orbital Motion:
   - Orbital velocity: v = √(GM/r)
   - Orbital period: T = 2π√(r³/GM)
   - Escape velocity: v_e = √(2GM/r)
   - Binding energy: E = -GMm/(2r)

4. Kepler's Laws:
   - First law: Planets move in ellipses
   - Second law: dA/dt = L/(2m) = constant
   - Third law: T² ∝ a³
   - Mathematical form: T² = (4π²/GM)a³

5. Variation of g:
   - With height: g(h) = g(0)/(1 + h/R)²
   - With depth: g(d) = g(0)(1 - d/R)
   - At Earth's surface: g = 9.8 m/s²

🔬 Properties of Matter

Properties of Matter Overview:

Properties of Matter
├── Elasticity
│   ├── Stress and Strain
│   ├── Hooke's Law
│   ├── Elastic Moduli
│   │   ├── Young's Modulus
│   │   ├── Bulk Modulus
│   │   ├── Shear Modulus
│   │   └── Modulus of Rigidity
│   ├── Elastic Potential Energy
│   └── Elastic Limit
├── Fluids
│   ├── Pressure in Fluids
│   ├── Buoyancy and Floatation
│   ├── Pascal's Law
│   ├── Archimedes' Principle
│   └── Fluid Dynamics
│       ├── Equation of Continuity
│       ├── Bernoulli's Equation
│       └── Applications
├── Surface Tension
│   ├── Surface Tension Force
│   ├── Surface Energy
│   ├── Capillarity
│   ├── Angle of Contact
│   └── Applications
├── Viscosity
│   ├── Viscous Force
│   ├── Poiseuille's Equation
│   ├── Stokes' Law
│   ├── Terminal Velocity
│   └── Reynolds Number
└── Mechanical Properties of Solids
    ├── Stress-Strain Relationships
    ├── Elastic Behavior
    ├── Plastic Deformation
    └── Fracture

Mechanical Properties Key Formulas:

Essential Properties of Matter Formulas:
1. Stress and Strain:
   - Stress: σ = F/A
   - Strain: ε = ΔL/L (linear strain)
   - Volumetric strain: ε_v = ΔV/V
   - Shear strain: γ = Δx/h

2. Elastic Moduli:
   - Young's modulus: Y = σ/ε = FL/(A·ΔL)
   - Bulk modulus: K = -P/ε_v = -P·ΔV/V
   - Shear modulus: η = F/(A·γ)
   - Relation: Y = 9KG/(3K + G)

3. Fluid Statics:
   - Pressure: P = ρgh
   - Buoyant force: F_b = ρ·V·g
   - Pascal's law: P = F/A (transmitted equally)

4. Bernoulli's Equation:
   - P + ½ρv² + ρgh = constant
   - Applications: Venturi meter, airplane wing

5. Viscosity:
   - Viscous force: F = ηA(dv/dy)
   - Terminal velocity: v_t = (2/9)(r²g/η)(ρ_s - ρ_f)
   - Poiseuille's equation: Q = πPr⁴/(8ηL)

6. Surface Tension:
   - Surface tension force: F = γL
   - Surface energy: U = γA
   - Capillary rise: h = 2γcosθ/(ρgr)

🎯 Problem-Solving Strategies

Mechanics Problem-Solving Framework:

Systematic Approach:
1. Understand the Problem
   - Identify the system
   - Determine given quantities
   - Identify what needs to be found
   - Note the constraints

2. Draw Free Body Diagrams
   - Isolate the object
   - Draw all forces
   - Use proper directions
   - Label all quantities

3. Apply Physical Laws
   - Newton's laws (F = ma)
   - Conservation laws
   - Kinematic equations
   - Energy principles

4. Solve Mathematically
   - Choose coordinate system
   - Resolve vectors if needed
   - Apply equations
   - Solve systematically

5. Check and Verify
   - Check units
   - Verify reasonableness
   - Consider special cases
   - Confirm physical sense

Common Problem Types:

Mechanics Problem Categories:
1. Kinematics Problems:
   - Projectile motion
   - Relative motion
   - Circular motion
   - Graphical analysis

2. Dynamics Problems:
   - Connected bodies
   - Inclined planes
   - Pulley systems
   - Circular dynamics

3. Energy Problems:
   - Work-energy theorem
   - Conservation of energy
   - Power calculations
   - Collision analysis

4. Rotational Problems:
   - Moment of inertia
   - Torque and angular acceleration
   - Angular momentum
   - Rolling motion

5. Gravitation Problems:
   - Orbital motion
   - Escape velocity
   - Gravitational potential
   - Kepler's laws

📈 Performance Tips

Exam Success Strategies:

• Master vector analysis for force problems
• Practice free body diagrams extensively
• Memorize all kinematic equations with conditions
• Understand energy conservation deeply
• Practice moment of inertia calculations
• Focus on conceptual understanding over rote memorization

Use this comprehensive mechanics mindmap to master all JEE Advanced mechanics concepts! Systematic practice with these visual aids will significantly enhance your problem-solving abilities. 🎯