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
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
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
Motion of System of Particles and Rigid Body
Laws of Motion
Work, Energy and Power
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
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)
Properties of Bulk Matter
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
Behaviour of Perfect Gases and Kinetic Theory of Gases
Mechanical Properties of Solids
 Elastic Behaviour of Solid
 Stress and Strain
 Hooke’s Law
 Stressstrain Curve
 Young’s Modulus
 Determination of Young’s Modulus of the Material of a Wire
 Shear Modulus or Modulus of Rigidity
 Bulk Modulus
 Application of Elastic Behaviour of Materials
 Elastic Energy
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
 Reflection of Transverse and Longitudinal Waves
 Displacement Relation for a Progressive Wave
 The Speed of a Travelling Wave
 Principle of Superposition of Waves
 Introduction of Reflection of Waves
 Standing Waves and Normal Modes
 Beats
 Doppler Effect
 Wave Motion
 Speed of Wave Motion
Notes
Potential energy of spring

The spring force is an example of a variable force, which is conservative.

In an ideal spring, Fs = − kx , this force law for the spring is called Hooke’s law.

The constant k is called the spring constant. Its unit is N m1.

The spring is said to be stiff if k is large and soft if k is small.
Spring force is position dependent as first stated by Hooke,
Fs = − kx

Work done by spring force only depends on the initial and final positions. Thus, the spring force is a conservative force.

We define the potential energy V(x) of the spring to be zero when block and spring system is in the equilibrium position.

If the extension is xm, the work done by the spring force is
`W_s = ∫_0^(x_m) F_s dx=∫_0^(x_m) kx dx`
`=(kx_m^2)/2`
The same is true when the spring is compressed with a displacement xc (< 0). The spring force does work Ws = kx2/2 while the external force F does work + kxc^2/2. If the block is moved from an initial displacement xi to a final displacement xf, the work done by the spring force Ws is
`W_s = ∫_(x_i)^(x_f) kx dx = (kx_i^2)/2  (kx_f^2)/2`
Thus the work done by the spring force depends only on the end points. Specifically, if the block is pulled from xi and allowed to return to xi;
`W_s = ∫_(x_i)^(x_f) kx dx = (kx_i^2)/2  (kx_f^2)/2 = 0`
The work done by the spring force in a cyclic process is zero.
We have explicitly demonstrated that the spring force
(i) is position dependent only as first stated by Hooke, (Fs = − kx);
(ii) does work which only depends on the initial and final positions.
Thus, the spring force is a conservative force.
We define the potential energy V(x) of the spring to be zero when block and spring system is in the equilibrium position. For an extension (or compression) x the above analysis suggests that
`V(x)=(kx^2)/2`
If the block of mass m is extended to xm and released from rest, then its total mechanical energy at any arbitrary point x (where x lies between – xm and + xm) will be given by:
`1/2 kx_m^2 = 1/2 kx^2 + 1/2 mv^2`
This suggests that the speed and the kinetic energy will be maximum at the equilibrium position, x = 0, i.e.,
`1/2 mv_m^2 = 1/2 kx_m^2`
where vm is the maximum speed.
Or `v_m= sqrt(k/m) x_m`
Note that k/m has the dimensions of `[T^(2)]` and our equation is dimensionally correct.
The kinetic energy gets converted to potential energy and vice versa, however, the total mechanical energy remains constant. This is graphically depicted in Fig
Example: A car of mass M travelling with speed v, collides with a spring, having spring constant k. The car comes to rest (momentarily) when spring is compressed to a distance of `x_m`. Find `x_m`
Solution: Work done on car = `K_fK_i=0(mv^2)/2 =(mv^2)/2`
Work done by spring, `W_s= ∫_(x_i)^(x_f) kx dx =∫_(0)^(x_m) kx dx = (kx_m^2)/2`
Now work done by spring equals work done on car => `(mv^2)/2 =  (kx_m^2)/2`
`x_m = sqrt (m/k) v`