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 and Least Count of Measuring Instruments
 Errors in Measurements
 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
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 and Its Characteristics
 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 of Motion in One Dimension
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 (Average and Instantaneous)
 Relative Velocity
 Elementary Concept of Differentiation and Integration for Describing Motion
 Uniform and Nonuniform Motion
 Uniformly Accelerated Motion
 Positiontime, Velocitytime and Accelerationtime Graphs
 Position  Time Graph
 Relations for Uniformly Accelerated Motion (Graphical Treatment)
 Introduction of Motion in One Dimension
Motion in a Plane
 Scalars and Vectors
 Multiplication of Vectors by a Real Number or Scalar
 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 (UCM)
 General Vectors and Their Notations
 Motion in a Plane  Average Velocity and Instantaneous Velocity
 Rectangular Components
 Scalar (Dot) and Vector (Cross) 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 of Motion in One Dimension
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
 Conservation of Mechanical Energy
 Potential Energy of a Spring
 Various Forms of Energy : the Law of Conservation of Energy
 Power
 Collisions
 Non  Conservative Forces  Motion in a Vertical Circle
Laws of Motion
Motion of System of Particles and Rigid Body
Gravitation
 Kepler’s Laws
 Newton’s Universal Law of Gravitation
 The Gravitational Constant
 Acceleration Due to Gravity of the Earth
 Acceleration Due to Gravity Below and Above the Earth's Surface
 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
Work, Energy and Power
Properties of Bulk Matter
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 Body
 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
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 Process
 Heat Engine
 Refrigerators and Heat Pumps
 Second Law of Thermodynamics
 Reversible and Irreversible Processes
 Carnot Engine
 Isothermal Processes
 Adiabatic Processes
Mechanical Properties of Solids
Behaviour of Perfect Gases and Kinetic Theory of Gases
Mechanical Properties of Fluids
 Thrust and Pressure
 Pascal’s Law
 Variation of Pressure with Depth
 Atmospheric Pressure and Gauge Pressure
 Hydraulic Machines
 Streamline and Turbulent Flow
 Applications of Bernoulli’s Equation
 Viscous Force or Viscosity
 Reynold's Number
 Surface Tension
 Effect of Gravity on Fluid Pressure
 Terminal Velocity
 Critical Velocity
 Excess of Pressure Across a Curved Surface
 Introduction of Mechanical Properties of Fluids
 Archimedes' Principle
 Stoke's Law
 Equation of Continuity
 Torricelli's Law
Oscillations and Waves
Thermal Properties of Matter
 Heat and Temperature
 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 Black Body Radiation
 Wien's Displacement Law
 Stefan's Law
 Anomalous Expansion of Water
 Liquids and Gases
 Thermal Expansion of Solids
 Green House Effect
Thermodynamics
Kinetic Theory
 Molecular Nature of Matter
 Gases and Its Characteristics
 Equation of State of a Perfect Gas
 Work Done in Compressing a Gas
 Introduction of Kinetic Theory of an Ideal Gas
 Interpretation of Temperature in Kinetic Theory
 Law of Equipartition of Energy
 Specific Heat Capacities  Gases
 Mean Free Path
 Kinetic Theory of Gases  Concept of Pressure
 Assumptions of Kinetic Theory of Gases
 RMS Speed of Gas Molecules
 Degrees of Freedom
 Avogadro's Number
Oscillations
 Periodic and Oscillatory Motion
 Simple Harmonic Motion (S.H.M.)
 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
 Accuracy of measuring instruments
 Precision of measuring instruments
 Least count for various instruments
 Zero error: Negative and Positive zero error
notes
Accuracy, Precision of Instruments and Errors in Measurements
Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity, which is based on measured values, also has an error.
Accuracy: The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.
Precision: Precision tells us to what resolution or limit the quantity is measured.
For example: If the true value of a certain length is 3.678 cm and two instruments with different resolutions, up to 1 (less precise) and 2 (more precise) decimal places respectively, are used. And if first measures the length as 3.5 and the second as 3.38 then the first has more accuracy but less precision while the second has less accuracy and more precision.
Thus every measurement is approximate due to errors in measurement.
In general, the errors in measurement can be broadly classified as follows:
(a) Systematic errors
(b) Random errors.
Systematic Error
The systematic errors are those errors that tend to be in one direction, either positive or negative. Errors due to buoyancy in weighting and radiation loss in calorimetry are systematic errors. They can eliminated by manipulation. Some of the sources of systematic errors are
1. Instrument errors: These arise from imperfect design or calibration error in the instrument. Worn off scale, zero error in a weighing scale are some examples of instrument errors.
2. Imperfections in experimental techniques: If the technique is not accurate (for example measuring temperature of human body by placing thermometer under armpit resulting in lower temperature than actual) and due to the external conditions like temperature, wind, humidity, these kinds of errors occur.
3. Personal errors: Errors occurring due to human carelessness, lack of proper setting, taking down incorrect reading are called personal errors
These errors can be removed by:

Taking proper instrument and calibrating it properly.
Experimenting under proper atmospheric conditions and techniques.
Random errors
 Errors which occur at random with respect to sign and size are called Random errors.
 These occur due to unpredictable fluctuations in experimental conditions like temperature, voltage supply, mechanical vibrations, personal errors etc.
Least Count Error
Smallest value that can be measured by the measuring instrument is called its least count. Least count error is the error associated with the resolution or the least count of the instrument.

Least count errors can be minimized by using instruments of higher precision/resolution and improving experimental techniques (taking several readings of a measurement and then taking a mean).

Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors.
For example, a vernier callipers has the least count as 0.01cm; a spherometer may have a least count of 0.001 cm.
Absolute Error, Relative Error and Percentage Error
Absolute Error:
If a1,a2,a3,....,an be the measured values of a quantity in several measurements, then their mean is considered to be the true values of that quantity i.e.,
true value a0 = amean = `[a_1 + a_2 + a_3 + ... +a_n ]/[n]`
The magnitude of the difference between the true value of the quantity and the individual measurements value is the absolute error of that measurement. Hence, absolute errors in measured values are:
`Δa_1 = a_0  a_1, Δa_2 = a_0  a_2, Δa_3 = a_0  a_3, ..........Δa_n = a_0  a_n`
The arithmetic mean(i.e., the mean of the magnitudes) of all the absolute error is known as mean absolute error.
Δamean = `[Δa_1 + Δa_2 + Δa_3 + ......+ Δa_n]/[n]`
Relative Error:
The ratio between mean absolute error and the mean values is called relative error
Relative error = `"Mean absolute error"/"Mean value"`
= `(triangle "a"_"mean")/("a"_"mean")` = `(triangle "a"_"mean")/("a"_"0")`
Percentage Error:
Percentage Error is the expression of the relative error in percentage.
Percentage Error = `(triangle "a"_"mean")/("a"_"mean") ` × 100%
Combination of errors
If a quantity depends on two or more other quantities, the combination of errors in the two quantities helps to determine and predict the errors in the resultant quantity. There are several procedures for this.
Suppose two quantities A and B have values as A ± ΔA and B ± ΔB. Z is the result and ΔZ is the error due to combination of A and B.
Criteria  Sum or Difference  Product  Raised to Power 
Resultant value Z  `Z=A+B`  `Z=AB`  `Z= A^k` 
Result with error  `Z+ ΔZ=(A+ΔA)+(B+ΔB)`  `Z+ ΔZ=(A+ΔA)(B+ΔB)`  `Z+ ΔZ=(A+ΔA)^k` 
Resultant error range  `+ ΔZ=+ΔA+ΔB`  `(ΔZ)/Z=(ΔA)/A+(ΔB)/B`  
Maximum error  `ΔZ=ΔA +ΔB`  `(ΔZ)/Z=(ΔA)/A+(ΔB)/B`  `(ΔZ)/Z=k((ΔA)/A)` 
Error  Sum of absolute errors  Sum of relative errors  k times relative error 