# Learn Vectors, Relativity and More with Introduction to Classical Mechanics by Takwale and Puranik

## Introduction To Classical Mechanics By Takwale Pdf.rar

Classical mechanics is one of the oldest and most fundamental branches of physics. It deals with the motion and interaction of particles and rigid bodies under the influence of forces. Classical mechanics has many applications in engineering, astronomy, chemistry, biology and other fields. It also provides the foundation for more advanced topics such as quantum mechanics, nuclear physics and electrodynamics.

## Introduction To Classical Mechanics By Takwale Pdf.rar

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If you are looking for a comprehensive and lucid introduction to classical mechanics, you might want to check out Introduction To Classical Mechanics By R.G. Takwale & P.S. Puranik. This book is a pdf file that can be downloaded for free from various online sources. In this article, we will give you an overview of what this book offers and why you should read it.

## About the book

The authors of this book are R.G. Takwale and P.S. Puranik, who are both professors of physics at Pune University in India. They have extensive experience in teaching classical mechanics to undergraduate students. They have also written several other books on physics topics such as optics, thermodynamics and electricity.

The book was first published in 1979 by Tata McGraw-Hill Education. It has been revised and updated several times since then. The latest edition was published in 2018 by Internet Archive. The book has 425 pages and contains 12 chapters. Each chapter has a summary, solved examples, exercises and references.

## Content overview

The book covers all the essential topics in classical mechanics that a student needs to know. It starts with a brief introduction to vectors, which are used to represent physical quantities such as displacement, velocity, acceleration and force. It then explains the basic principles of Newtonian mechanics, which describe how objects move under different types of forces. It also introduces the concepts of work, energy, power and conservation of energy.

The book then moves on to more advanced topics such as momentum, collisions, rotational motion, oscillations and gravitation. It shows how these concepts can be applied to various physical systems such as planets, satellites, pendulums and springs. It also discusses some special cases such as circular motion, central force motion and non-inertial frames.

The book also covers one of the most important topics in modern physics: the special theory of relativity. It explains how classical mechanics fails to account for some phenomena that occur at high speeds or near massive objects. It introduces the concepts of inertial frames, Lorentz transformations, time dilation, length contraction and mass-energy equivalence.

principle of least action, which states that the path of a system is such that the action is minimized.

## Vectors

Vectors are mathematical objects that have both magnitude and direction. They can be used to describe physical quantities such as displacement, velocity, acceleration and force. For example, if a car moves 10 km east and then 5 km north, its displacement can be represented by a vector that has a magnitude of 11.18 km and a direction of 26.57 degrees north of east.

Vectors can be added, subtracted, multiplied and divided by scalars (numbers) using simple rules. They can also be multiplied by other vectors using two operations: dot product and cross product. The dot product of two vectors gives a scalar that measures the projection of one vector onto another. The cross product of two vectors gives a vector that is perpendicular to both vectors and has a magnitude that measures the area of the parallelogram formed by them.

Vectors can be represented in different ways, such as by their components, by their magnitude and direction, or by their unit vectors. A unit vector is a vector that has a magnitude of one and points in a specific direction. For example, the unit vectors i, j and k point along the x, y and z axes respectively. Any vector can be expressed as a linear combination of unit vectors.

## Newtonian mechanics

Newtonian mechanics is the branch of classical mechanics that is based on the laws of motion formulated by Isaac Newton in the 17th century. These laws state that:

The first law: An object at rest remains at rest and an object in motion remains in motion with constant velocity unless acted upon by an external force.

The second law: The net force acting on an object is equal to its mass times its acceleration.

The third law: For every action, there is an equal and opposite reaction.

These laws can be used to analyze the motion of any object under the influence of forces. For example, if a ball is thrown upward with an initial velocity of 20 m/s, we can use the second law to find its acceleration (which is equal to -9.8 m/s^2 due to gravity), its velocity at any time (which decreases as it goes up and increases as it comes down), and its maximum height (which is 20.4 m).

Newtonian mechanics also introduces the concept of inertial frames of reference, which are frames in which the laws of motion hold true. An inertial frame is one that is either at rest or moving with constant velocity relative to another inertial frame. For example, a person standing on the ground and a person sitting in a train moving with constant speed are both in inertial frames. However, a person in a car that is accelerating or turning is not in an inertial frame.

## Work and energy

Work and energy are two related concepts that measure the ability of a force to cause motion or change. Work is defined as the product of force and displacement along the direction of force. For example, if a person pushes a box with a force of 50 N for a distance of 2 m along the floor, he does 100 J of work on the box.

Energy is defined as the capacity to do work. There are different forms of energy, such as kinetic energy (the energy of motion), potential energy (the energy stored due to position or configuration), thermal energy (the energy due to temperature), chemical energy (the energy due to chemical bonds), nuclear energy (the energy due to nuclear reactions), etc. Energy can be converted from one form to another, but it cannot be created or destroyed. This is known as the principle of conservation of energy.

One way to apply the principle of conservation of energy is by using the concept of mechanical energy, which is the sum of kinetic and potential energies. If there are no non-conservative forces (such as friction or air resistance) acting on a system, then the mechanical energy of the system remains constant. For example, if a pendulum swings back and forth without any damping, then its mechanical energy (the sum of its kinetic and potential energies) stays the same at any point.

## Momentum and collisions

Momentum is another physical quantity that measures the motion of an object. It is defined as the product of mass and velocity. For example, if a car has a mass of 1000 kg and moves with a velocity of 20 m/s eastward, its momentum is 20000 kg m/s eastward.

of conservation of momentum states that the total momentum of an isolated system (a system that does not interact with its surroundings) remains constant. For example, if two cars collide and stick together, their total momentum before and after the collision is the same.

The impulse of a force is defined as the product of force and time. It measures the change in momentum of an object due to a force. For example, if a ball is hit by a bat with a force of 100 N for 0.1 s, the impulse of the bat on the ball is 10 N s. This impulse causes the ball to change its momentum by 10 kg m/s.

Collisions are interactions between two or more objects that involve exchange of momentum and energy. There are different types of collisions, such as elastic collisions (where both momentum and kinetic energy are conserved), inelastic collisions (where only momentum is conserved and some kinetic energy is lost), and perfectly inelastic collisions (where the objects stick together and have the same final velocity).

## Rotational motion

Rotational motion is the motion of an object around a fixed axis or point. It involves concepts such as angular velocity, angular acceleration, torque, moment of inertia and angular momentum.

Angular velocity is the rate of change of angular displacement. It measures how fast an object rotates. For example, if a wheel rotates 90 degrees in one second, its angular velocity is 90 degrees per second or pi/2 radians per second.

Angular acceleration is the rate of change of angular velocity. It measures how fast an object changes its rotational speed. For example, if a wheel increases its angular velocity from 0 to pi radians per second in two seconds, its angular acceleration is pi/2 radians per second squared.

Torque is the rotational equivalent of force. It measures how much a force causes an object to rotate. It is defined as the product of force and perpendicular distance from the axis of rotation. For example, if a person applies a force of 10 N at a distance of 0.5 m from the hinge of a door, the torque on the door is 5 N m.

Moment of inertia is the rotational equivalent of mass. It measures how much an object resists rotational motion. It depends on the mass distribution and shape of the object. For example, a solid sphere has a smaller moment of inertia than a hollow sphere with the same mass and radius.

Angular momentum is the rotational equivalent of linear momentum. It measures the amount of rotational motion of an object. It is defined as the product of moment of inertia and angular velocity. For example, if a solid sphere has a mass of 2 kg, a radius of 0.1 m and an angular velocity of pi radians per second, its angular momentum is 0.02 kg m^2/s.

of angular momentum states that the total angular momentum of an isolated system (a system that does not interact with its surroundings) remains constant. For example, if a spinning top is placed on a smooth table, its angular momentum stays the same unless an external torque is applied.

The torque on an object is equal to the rate of change of its angular momentum. For example, if a person applies a torque of 5 N m to a wheel with a moment of inertia of 10 kg m^2, the wheel's angular momentum changes by 0.5 kg m^2/s every second.

## Oscillations

Oscillations are periodic motions that repeat themselves after a fixed interval of time. They involve concepts such as simple harmonic motion, damped oscillations, forced oscillations and resonance.

Simple harmonic motion is the simplest type of oscillation. It occurs when an object moves back and forth around an equilibrium position under the influence of a restoring force that is proportional to the displacement. For example, a mass attached to a spring or a pendulum swinging in a small angle undergo simple harmonic motion.

Simple harmonic motion can be described by four parameters: amplitude, frequency, period and phase. Amplitude is the maximum displacement from the equilibrium position. Frequency is the number of oscillations per unit time. Period is the time taken for one complete oscillation. Phase is the angle that measures the initial position of the object relative to the equilibrium position.

Damped oscillations are oscillations that gradually lose energy due to friction or other dissipative forces. They have a decreasing amplitude and frequency over time. For example, a pendulum swinging in air or a mass-spring system with friction undergo damped oscillations.

Forced oscillations are oscillations that are driven by an external periodic force. They have a constant amplitude and frequency that depend on the characteristics of the system and the driving force. For example, a mass-spring system attached to a vibrating motor or a tuning fork struck by a hammer undergo forced oscillations.

Resonance is a phenomenon that occurs when the frequency of the driving force matches the natural frequency of the system. It results in a large amplitude and energy transfer. For example, a glass shattering when exposed to a loud sound or a bridge collapsing when subjected to wind vibrations are examples of resonance.

## Gravitation

Gravitation is the universal force of attraction between any two objects that have mass. It involves concepts such as gravitational force, potential energy, escape velocity and Kepler's laws.

Gravitational force is the force that acts between any two objects due to their masses. It is proportional to the product of their masses and inversely proportional to the square of their distance. It also acts along the line joining their centers. For example, the gravitational force between the earth and the moon is about 2 x 10^20 N.

the earth's surface is mgh, where m is the mass of the object, g is the acceleration due to gravity and h is the height.

Escape velocity is the minimum speed required for an object to escape from the gravitational field of a planet or a star. It is equal to the square root of twice the product of the gravitational constant and the mass of the planet or star divided by its radius. For example, the escape velocity from the earth's surface is about 11.2 km/s.

Kepler's laws are three empirical laws that describe the motion of planets around the sun. They state that:

The first law: The orbit of each planet is an ellipse with the sun at one of its foci.

The second law: The line joining a planet and the sun sweeps out equal areas in equal intervals of time.

The third law: The square of the orbital period of a planet is proportional to the cube of its average distance from the sun.

## Special theory of relativity

Special theory of relativity is a branch of modern physics that deals with the phenomena that occur when objects move at speeds close to the speed of light. It involves concepts such as inertial frames, Lorentz transformations, time dilation, length contraction and mass-energy equivalence.

Inertial frames are frames of reference in which the laws of physics hold true and are the same for all observers. They are either at rest or moving with constant velocity relative to each other. For example, a person standing on the ground and a person sitting in a spaceship moving with constant speed are both in inertial frames.

Lorentz transformations are mathematical equations that relate the space and time coordinates of events as measured by different inertial observers. They show how space and time are not absolute but relative to the motion of the observer. For example, if a spaceship moves with a speed of 0.8c (where c is the speed of light) relative to an observer on earth, then according to Lorentz transformations:

The length of the spaceship as measured by the observer on earth is 60% of its length as measured by an observer on the spaceship.

The time interval between two events on the spaceship as measured by the observer on earth is 67% longer than the time interval as measured by an observer on the spaceship.

The simultaneity of two events on the spaceship as measured by the observer on earth depends on their position along the direction of motion.

the moving object, t is the time interval as measured by the observer at rest, v is the speed of the object and c is the speed of light. For example, if a clock on a spaceship moves with a speed of 0.8c relative to an observer on earth, then one second on the clock corresponds to 1.67 seconds on the earth.

Length contraction is the phenomenon that length shrinks for an object moving at high speed relative to an observer at rest. It is also a consequence of Lorentz transformations and can be calculated using the formula: l' = l*sqrt(1-v^2/c^2), where l' is the length as measured by the moving object, l is the length as measured by the observer at rest, v is the speed of the object and c is the speed of light. For example, if a meter stick on a spaceship moves with a speed of 0.8c relative to an observer on earth, then its length as measured by the observer on earth is 0.6 m.

Mass-energy equivalence is the principle that mass and energy are equivalent and can be converted into each other. It can be expressed by the famous equation: E = mc^2, where E is the energy, m is the mass and c is the speed of light. For example, if a mass of 1 kg is converted into energy, it produces 9 x 10^16 J of energy, which is equivalent to the energy released by about 21 megatons of TNT.

## Lagrangian and Hamiltonian mechanics

Lagrangian and Hamiltonian mechanics are alternative formulations of classical mechanics that use generalized coordinates instead of Cartesian coordinates to describe the motion of a system. They involve concepts such as generalized coordinates, Lagrange's equations, Hamilton's principle and Hamilton's equations.

Generalized coordinates are coordinates that can be chosen arbitrarily to describe the configuration of a system. They can be angles, lengths, areas or any other quantities that specify the position and orientation of the system. For example, for a simple pendulum, we can use either the Cartesian coordinates (x,y) or the polar coordinates (r,theta) as generalized coordinates.

Lagrange's equations are differential equations that relate the generalized coordinates and their derivatives to the forces acting on the system. They can be derived from a function called the Lagrangian, which is defined as the difference between the kinetic and potential energies of the system. For example, for a simple pendulum with mass m and length l, the Lagrangian is L = (1/2)m(l^2)(dtheta/dt)^2 - mglcos(theta), and Lagrange's equation is d/dt(dL/d(dtheta/dt)) - dL/dtheta = 0.

Hamilton's principle is a variational principle that states that the path of a system in configuration space is such that the action is minimized. The action is defined as the integral of the Lagrangian over time. For example, for a simple pendulum with mass m and length l, the action is S = integral(L dt) = integral((1/2)m(l^2)(dtheta/dt)^2 - mglcos(theta) dt).

the generalized coordinates and their conjugate momenta to the forces acting on the system. They can be derived from a function called the Hamiltonian, which is defined as the sum of the kinetic and potential energies of the system. For example, for a simple pendulum with mass m and length l, the Hamiltonian is H = (1/2)m(l^2)(dtheta/dt)^2 + mglcos(theta), and Hamilton's equations are dtheta/dt = dH/dp and dp/dt = -dH/dtheta, where p is the conjugate momentum of theta.

## Conclusion

In this article, we have given you a brief overview of Introduction To Classical Mechanics By R.G. Takwale & P.S. Puranik, a pdf file that can be downloaded for free from various online sources. This book is a lucid and comprehensive introduction to vectors, classical mechanics and special theory of relativity. It covers all the essential topics that a student needs to know, such as Newton's laws, work and energy, momentum and collisions, rotational motion, oscillations, gravitation, Lorentz transformations, Lagrangian and Hamiltonian mechanics and more. It also provides summaries, solved examples, exercises and references for each chapter.

If you are interested in learning classical mechanics, this book is a great resource for you. It will give you a solid foundation for further studies in physics and other related fields. It will also help you develop your analytical and problem-solving ski