Topics
Physical World and Measurement
Physical World
Units and Measurements
 International System of Units
 Measurement of Length
 Measurement of Mass
 Measurement of Time
 Accuracy Precision of Instruments and Errors in Measurement
 Significant Figures
 Dimensions of Physical Quantities
 Dimensional Formulae and Dimensional Equations
 Dimensional Analysis and Its Applications
 Need for Measurement
 Units of Measurement
 Fundamental and Derived Units
 Length, Mass and Time Measurements
 Introduction of Units and Measurements
Kinematics
Motion in a Plane
 Scalars and Vectors
 Multiplication of Vectors by a Real Number
 Addition and Subtraction of Vectors — Graphical Method
 Resolution of Vectors
 Vector Addition – Analytical Method
 Motion in a Plane
 Motion in a Plane with Constant Acceleration
 Projectile Motion
 Uniform Circular Motion
 General Vectors and Their Notations
 Motion in a Plane  Average Velocity and Instantaneous Velocity
 Rectangular Components
 Scalar and Vector Product of Vectors
 Relative Velocity in Two Dimensions
 Cases of Uniform Velocity
 Cases of Uniform Acceleration Projectile Motion
 Motion in a Plane  Average Acceleration and Instantaneous Acceleration
 Angular Velocity
 Introduction
Motion in a Straight Line
 Position, Path Length and Displacement
 Average Velocity and Average Speed
 Instantaneous Velocity and Speed
 Kinematic Equations for Uniformly Accelerated Motion
 Acceleration
 Relative Velocity
 Elementary Concepts of Differentiation and Integration for Describing Motion
 Uniform and NonUniform Motion
 Uniformly Accelerated Motion
 Positiontime, Velocitytime and Accelerationtime Graphs
 Motion in a Straight Line  Positiontime Graph
 Relations for Uniformly Accelerated Motion (Graphical Treatment)
 Introduction
Laws of Motion
 Aristotle’s Fallacy
 The Law of Inertia
 Newton'S First Law of Motion
 Newton’s Second Law of Motion
 Newton's Third Law of Motion
 Conservation of Momentum
 Equilibrium of a Particle
 Common Forces in Mechanics
 Circular Motion
 Solving Problems in Mechanics
 Static and Kinetic Friction
 Laws of Friction
 Inertia
 Intuitive Concept of Force
 Dynamics of Uniform Circular Motion  Centripetal Force
 Examples of Circular Motion (Vehicle on a Level Circular Road, Vehicle on a Banked Road)
 Lubrication  (Laws of Motion)
 Law of Conservation of Linear Momentum and Its Applications
 Rolling Friction
 Introduction
Work, Energy and Power
 Introduction of Work, Energy and Power
 Notions of Work and Kinetic Energy: the WorkEnergy Theorem
 Kinetic Energy
 Work Done by a Constant Force and a Variable Force
 Concept of Work
 The Concept of Potential Energy
 The Conservation of Mechanical Energy
 Potential Energy of a Spring
 Various Forms of Energy : the Law of Conservation of Energy
 Power
 Concept of Collisions
 Non  Conservative Forces  Motion in a Vertical Circle
Motion of System of Particles and Rigid Body
System of Particles and Rotational Motion
 Motion  Rigid Body
 Centre of Mass
 Motion of Centre of Mass
 Linear Momentum of a System of Particles
 Vector Product of Two Vectors
 Angular Velocity and Its Relation with Linear Velocity
 Torque and Angular Momentum
 Equilibrium of Rigid Bodies
 Moment of Inertia
 Theorems of Perpendicular and Parallel Axes
 Kinematics of Rotational Motion About a Fixed Axis
 Dynamics of Rotational Motion About a Fixed Axis
 Angular Momentum in Case of Rotation About a Fixed Axis
 Rolling Motion
 Momentum Conservation and Centre of Mass Motion
 Centre of Mass of a Rigid Body
 Centre of Mass of a Uniform Rod
 Rigid Body Rotation
 Equations of Rotational Motion
 Comparison of Linear and Rotational Motions
 Values of Moments of Inertia for Simple Geometrical Objects (No Derivation)
Gravitation
 Kepler’S Laws
 Universal Law of Gravitation
 The Gravitational Constant
 Acceleration Due to Gravity of the Earth
 Acceleration Due to Gravity Below and Above the Surface of Earth
 Acceleration Due to Gravity and Its Variation with Altitude and Depth
 Gravitational Potential Energy
 Escape Speed
 Earth Satellites
 Energy of an Orbiting Satellite
 Geostationary and Polar Satellites
 Weightlessness
 Escape Velocity
 Orbital Velocity of a Satellite
Properties of Bulk Matter
Mechanical Properties of Fluids
 Concept of Pressure
 Pascal's Law
 Variation of Pressure with Depth
 Atmospheric Pressure and Gauge Pressure
 Hydraulic Machines
 STREAMLINE FLOW
 Bernoulli’S Principle
 Viscosity
 Reynolds Number
 Surface Tension
 Effect of Gravity on Fluid Pressure
 Terminal Velocity
 Critical Velocity
 Excess of Pressure Across a Curved Surface
 Introduction to Fluid Machanics
 Archimedes' Principle
 Stokes' Law
 Equation of Continuity
 Torricelli'S Law
Thermal Properties of Matter
 Temperature and Heat
 Measurement of Temperature
 Idealgas Equation and Absolute Temperature
 Thermal Expansion
 Specific Heat Capacity
 Calorimetry
 Change of State  Latent Heat Capacity
 Conduction
 Convection
 Radiation
 Newton’s Law of Cooling
 Qualitative Ideas of Blackbody Radiation
 Wein'S Displacement Law
 Stefan's Law
 Anomalous Expansion of Water
 Liquids and Gases
 Thermal Expansion of Solids
 Green House Effect
Mechanical Properties of Solids
Thermodynamics
 Thermal Equilibrium
 Zeroth Law of Thermodynamics
 Heat, Internal Energy and Work
 First Law of Thermodynamics
 Specific Heat Capacity
 Thermodynamic State Variables and Equation of State
 Thermodynamic Processes
 Heat Engines
 Refrigerators and Heat Pumps
 Second Law of Thermodynamics
 Reversible and Irreversible Processes
 Carnot Engine
 Isothermal Processes
 Adiabatic Processes
Behaviour of Perfect Gases and Kinetic Theory of Gases
Kinetic Theory
 Molecular Nature of Matter
 Behaviour of Gases
 Equation of State of a Perfect Gas
 Work Done in Compressing a Gas
 Introduction of Kinetic Theory of an Ideal Gas
 Kinetic Interpretation of Temperature
 Law of Equipartition of Energy
 Specific Heat Capacities  Gases
 Mean Free Path
 Kinetic Theory of Gases  Concept of Pressure
 Kinetic Theory of Gases Assumptions
 rms Speed of Gas Molecules
 Degrees of Freedom
 Avogadro's Number
Oscillations and Waves
Oscillations
 Periodic and Oscillatory Motions
 Simple Harmonic Motion
 Simple Harmonic Motion and Uniform Circular Motion
 Velocity and Acceleration in Simple Harmonic Motion
 Force Law for Simple Harmonic Motion
 Energy in Simple Harmonic Motion
 Some Systems Executing Simple Harmonic Motion
 Damped Simple Harmonic Motion
 Forced Oscillations and Resonance
 Displacement as a Function of Time
 Periodic Functions
 Oscillations  Frequency
Waves
description
 Checking the Dimensional Consistency of Equations
 Deducing Relation among the Physical Quantities
notes
Dimensional Analysis and its Applications
Dimensional Analysis:

Only those physical quantities which have same dimensions can be added and subtracted. This is called principle of homogeneity of dimensions.

Dimensions can be multiplied and cancelled like normal algebraic methods.

In mathematical equations, quantities on both sides must always have same dimensions.

Arguments of special functions like trigonometric, logarithmic and ratio of similar physical

Quantities are dimensionless.
Equations are uncertain to the extent of dimensionless quantities.
Example Distance = Speed x Time. In Dimension terms, [L] = [LT1] x [T]
Since, dimensions can be cancelled like algebra, dimension [T] gets cancelled and the equation becomes [L] = [L].
Homogeneity Principle
If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle.
Mathematically [LHS] = [RHS]
Limitations of Dimensional Analysis
1. Dimensionless quantities cannot be determined by this method. Constant of proportionality cannot be determined by this method. They can be found either by experiment (or) by theory.
2. This method is not applicable to trigonometric, logarithmic and exponential functions.
3. In the case of physical quantities which are dependent upon more than three physical quantities, this method will be difficult.
4. In some cases, the constant of proportionality also possesses dimensions. In such cases, we cannot use this system.
5. If one side of the equation contains addition or subtraction of physical quantities, we cannot use this method to derive the expression.
Applications:
Dimensional analysis is very important when dealing with physical quantities. In this section, we will learn about some applications of the dimensional analysis.
Fourier laid down the foundations of dimensional analysis. The Dimensional formulas are used to:
1. Verify the correctness of a physical equation.
2. Derive a relationship between physical quantities.
3. Converting the units of a physical quantity from one system to another system.
Checking the Dimensional Consistency
As we know, only similar physical quantities can be added or subtracted, thus two quantities having different dimensions cannot be added together. For example, we cannot add mass and force or electric potential and resistance.
For any given equation, the principle of homogeneity of dimensions is used to check the correctness and consistency of the equation. The dimensions of each component on either side of the sign of equality are checked, and if they are not the same, the equation is considered wrong.
Let us consider the equation given below,
`1/2`mv2 = mgh
The dimensions of the LHS and the RHS are calculated
LHS: [M] [LT1]2 = [M][L2T2] = [ML2T2]
RHS: [M] [LT2][L] = [ML2T2]
As we can see the dimensions of the LHS and the RHS are the same, hence, the equation is consistent.
Deducing the Relation among Physical Quantities
Dimensional analysis is also used to deduce the relation between two or more physical quantities. If we know the degree of dependence of a physical quantity on another, that is the degree to which one quantity changes with the change in another, we can use the principle of consistency of two expressions to find the equation relating these two quantities. This can be understood more easily through the following illustration.
Example: Derive the formula for centripetal force F acting on a particle moving in a uniform circle.
As we know, the centripetal force acting on a particle moving in a uniform circle depends on its mass m, velocity v and the radius r of the circle. Hence, we can write
F = ma vb rc
Writing the dimensions of these quantities
[MLT2] = Ma[LT1]bLc
[MLT2] = MaLc+bTb
As per the principle of homogeneity, we can write,
a = 1, b + c = 1 and b = 2
Solving the above three equations we get, a = 1, b = 2 and c = 1.
Hence, the centripetal force F can be represented as,
F = Km1v2r1
F = k`"mv"^2/"r"`