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
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 Non-uniform Motion
- Uniformly Accelerated Motion
- Position-time, Velocity-time and Acceleration-time Graphs
- Position - Time Graph
- Relations for Uniformly Accelerated Motion (Graphical Treatment)
- Introduction of Motion in One Dimension
- Motion in a Straight Line
Kinematics
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
Laws of Motion
Work, Energy and Power
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
Work, Energy and Power
- Introduction of Work, Energy and Power
- Notions of Work and Kinetic Energy: the Work-energy Theorem
- Kinetic Energy (K)
- Work Done by a Constant Force and a Variable Force
- Concept of Work
- Potential Energy (U)
- Conservation of Mechanical Energy
- Potential Energy of a Spring
- Various Forms of Energy : the Law of Conservation of Energy
- Power
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- 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 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
Gravitation
- Kepler’s Laws
- Newton’s Universal Law of Gravitation
- The Gravitational Constant
- Acceleration Due to Gravity of the Earth
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Properties of Bulk Matter
Mechanical Properties of Solids
- Elastic Behaviour of Solid
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- Hooke’s Law
- Stress-strain Curve
- Young’s Modulus
- Determination of Young’s Modulus of the Material of a Wire
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- Bulk Modulus
- Application of Elastic Behaviour of Materials
- Elastic Energy
- Poisson’s Ratio
Thermodynamics
Behaviour of Perfect Gases and Kinetic Theory of Gases
Mechanical Properties of Fluids
- Thrust and Pressure
- Pascal’s Law
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- Atmospheric Pressure and Gauge Pressure
- Hydraulic Machines
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- Applications of Bernoulli’s Equation
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- Reynold's Number
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- Terminal Velocity
- Critical Velocity
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- 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
- Ideal-gas Equation and Absolute Temperature
- Thermal Expansion
- Specific Heat Capacity
- Calorimetry
- Change of State - Latent Heat Capacity
- Conduction
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- Newton’s Law of Cooling
- Qualitative Ideas of Black Body Radiation
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- Liquids and Gases
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- Green House Effect
Thermodynamics
- Thermal Equilibrium
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- Heat Engine
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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
- Simple Pendulum
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
Motion of a car on a level road
Three forces act on the car:
(i) The weight of the car, mg
(ii) Normal reaction, N
(iii) Frictional force, f
As there is no acceleration in the vertical direction
N – mg = 0
N = mg
The centripetal force required for circular motion is along the surface of the road, and is provided by the component of the contact force between road and the car tyres along the surface. This by definition is the frictional force
f =< μs N
According to the second law, the force f providing this acceleration is:
`f_c=(mv^2)/R`
hence, `(mv^2)/r =< μs N`
`v^2 =< (μs N R)/m`
`= μs m g R/m`
`= μs R g`
`v =< sqrt (μs R g)`
This is the maximum speed of a car in circular motion on a level road
Motion of a car on a banked road
Since there is no acceleration along the vertical direction, the net force along this direction must be zero. Hence,
In the vertical direction (Y axis)
`N cos ϴ = f sin ϴ + mg` ----------------- (i)
The centripetal force is provided by the horizontal components of N and f.
In horizontal direction (X axis)
`f cos ϴ + N sin ϴ = (mv^2)/r ` ------------- (ii)
Since we know that `f = μs N`
For maximum velocity, `f = μs N`
(i) becomes:
`N cos ϴ = μs N sin ϴ + mg`
Or, `N cos ϴ – μs N sin ϴ = mg`
Or, `N = (mg)/(cos ϴ- μs sin ϴ)`
Put the above value of N in (ii)
`μ s N cos ϴ + N sin ϴ = (mv^2)/r`
`(μs m g cos ϴ)/(cos ϴ- μs sin ϴ) + (mg sin ϴ)/(cos ϴ- μs sin ϴ) = (mv^2)/r`
`(mg (sin ϴ + μs cos ϴ))/ (cos ϴ – μs sin ϴ) = (mv^2)/r`
Divide the Numerator & Denominator by cos ϴ, we get
`v^2= (R g (tan ϴ +μs))/(1- μs tan ϴ)`
`v = sqrt(( R g (tan ϴ + μs))/(1- μs tan ϴ))`
This is the maximum speed of a car on a banked road.
Special case:
When the velocity of the car = `v_0`,
-
No f is needed to provide the centripetal force. (`μ_s =0`)
-
Little wear & tear of tyres take place.
`v_o = sqrt(R g (tan ϴ))`
Problem: A circular racetrack of radius 300 m is banked at an angle of 15°. If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the
(a) optimum speed of the race car to avoid wear and tear on its tyres, and
(b) maximum permissible speed to avoid slipping?
Solution.
R = 300m
ϴ = `15^o`
μs = 0.2
`v_o = sqrt (Rg tan ϴ) = sqrt(300 * 9.8 * tan 15^o) = 28.1` m/s
R = 30
`vmax = sqrt ((R g (tan ϴ + μs)) /(1- μs tan ϴ))`
`= sqrt(( 300 * 9.8 * (0.2 + tan 15o))/(1-0.2tan15o))`
`= 38.1 `m/s
