Chapter 25 · Class 11-12 & JEE/NEET
Units, Dimensions & Basic Maths
47 topics in 12 parts
General dimensional formula Dimensions of force Dimensions of energy Dimensions of pressure Principle of dimensional homogeneity Unit conversion by dimensions
Key ideas
4. Systematic Comparison & Parameter Classifications
| Body Shape | Axis of Rotation | Moment of Inertia (I) | Shape Factor (k² / R²) |
|---|---|---|---|
| Thin Ring / Hollow Cylinder | Central Axis Perpendicular to Plane | M R² | 1.0 (Slowest Down Incline) |
| Solid Disc / Solid Cylinder | Central Axis Perpendicular to Plane | (1/2) M R² | 0.50 |
| Solid Sphere | Diameter Axis | (2/5) M R² | 0.40 (Fastest Down Incline) |
| Spherical Shell (Hollow) | Diameter Axis | (2/3) M R² | 0.67 |
| Uniform Rod (Length L) | Through Center Perpendicular to Length | (1/12) M L² | L² / (12 R²) |
Exam Worked Example
Problem: A solid sphere and a solid cylinder of equal mass M and radius R roll down the same inclined plane without slipping from rest. Find the ratio of their accelerations down the incline.
● Step 1: Formula for linear acceleration in pure rolling: a = (g sin θ) / (1 + k²/R²).
● Step 2: For solid sphere, k²/R² = 2/5 = 0.4. Therefore, a_sphere = (g sin θ) / (1 + 2/5) = 5/7 g sin θ ≈ 0.714 g sin θ.
● Step 3: For solid cylinder, k²/R² = 1/2 = 0.5. Therefore, a_cyl = (g sin θ) / (1 + 1/2) = 2/3 g sin θ ≈ 0.667 g sin θ.
● Step 4: Ratio a_sphere / a_cyl = (5/7) / (2/3) = (5 * 3) / (7 * 2) = 15 / 14 ≈ 1.07. The solid sphere reaches the bottom first!
Checkpoint
1. When a ballet dancer pulls her arms inward during a spin on smooth ice, what happens to her kinetic energy?
All 47 topics in Units, Dimensions & Basic Maths
Physical Quantities & Units
- Fundamental & Derived Quantities
- Fundamental & Derived Units
- Systems of Units & Unit Conversion
- Types of Physical Quantities
Dimensional Analysis
- Dimensions of Physical Quantities
- Dimensional Formulae & Equations
- Applications of Dimensional Analysis
- Limitations of Dimensional Analysis
Errors of Measurement & Error Analysis
- Absolute, Relative & Percentage Errors
- Propagation of Errors
- Precision & Accuracy
Significant Figures
- Rules for Significant Figures
- Rounding Off
- Calculations with Significant Figures
- Order of Magnitude
Vernier Callipers & Screw Gauge
- Reading a Vernier Callipers
- Constants & Reading of Screw Gauge
- Zero Error & Correction
- Searle's Method & Lab Experiments
Elementary Algebra & Functions
- Binomial Expression & Theorem
- Polynomial, Linear & Quadratic Equations
- Functions & Graph Analysis
Basic Trigonometry
- Trigonometric Ratios & Allied Angles
- Important Trigonometric Formulae
- Inverse Trigonometric Functions
- Sine & Cosine Rules
Differentiation
- Basic Formulae & Rules of Differentiation
- Chain, Product & Quotient Rules
- Derivative as Rate of Change
- Maxima & Minima
Integration
- Standard Formulae for Integration
- Rule of Substitution
- Definite Integral & Properties
- Area Under Curve
Vector Addition & Subtraction
- Representation & Types of Vectors
- Triangle & Parallelogram Laws
- Polygon Law of Vector Addition
- Analytical Subtraction of Vectors
- Vector Equilibrium
Components & Resolution of Vectors
- Rectangular Components in 2D & 3D
- Direction Cosines
- Unit Vectors & Position Vector
- Displacement Vector
Vector Products
- Dot Product of Vectors
- Cross Product of Vectors
- Angle Between Vectors
- Projection of Vectors