Topics
Physical World and Measurement
Physical World
Units and Measurements
 International System of Units
 Measurement of Length
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 Measurement of Time
 Accuracy Precision of Instruments and Errors in Measurement
 Significant Figures
 Dimensions of Physical Quantities
 Dimensional Formulae and Dimensional Equations
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 Need for Measurement
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 Fundamental and Derived Units
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 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
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 Projectile Motion
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 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 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
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
 Collisions
 Non  Conservative Forces  Motion in a Vertical Circle
Motion of System of Particles and Rigid Body
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 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
 Newton’s 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
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 Escape Velocity
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Properties of Bulk Matter
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 Thrust and Pressure
 Transmission of Pressure in Liquids: Pascal’s Law
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 STREAMLINE FLOW
 Applications of Bernoulli’s Equation
 Viscous Force Or Viscosity
 Reynolds Number
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 Introduction to Fluid Machanics
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 Stokes' Law
 Equation of Continuity
 Torricelli'S Law
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 Stefan's Law
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 Thermal Equilibrium
 Zeroth Law of Thermodynamics
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 First Law of Thermodynamics
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 Heat Engines
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 Carnot Engine
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Behaviour of Perfect Gases and Kinetic Theory of Gases
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
 Kinetic Theory of Gases Assumptions
 rms Speed of Gas Molecules
 Degrees of Freedom
 Avogadro's Number
Oscillations and Waves
Oscillations
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 Simple Harmonic Motion (SHM)
 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
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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`