Learn Classical Mechanics with Gupta Kumar Sharma: A PDF Book with Clear Explanations and Numerous Examples
Classical Mechanics By Gupta Kumar Sharma.pdf: A Comprehensive Review
Classical mechanics is one of the oldest and most fundamental branches of physics. It deals with the motion and interaction of particles and bodies under the influence of forces. It is also the basis for many other fields of physics, such as electromagnetism, thermodynamics, relativity, quantum mechanics, and cosmology.
Classical Mechanics By Gupta Kumar Sharma.pdf
But how can you learn classical mechanics in a systematic and effective way? What are the best books to read on this topic? And what are the advantages and disadvantages of each book?
In this article, I'm going to review one of the most popular books on classical mechanics: Classical Mechanics By Gupta Kumar Sharma.pdf. This book is written by two Indian authors, R.K. Gupta and S.L. Kumar, who are both professors of physics at Delhi University. The book was first published in 1995 by Pragati Prakashan, and has since been revised and updated several times. The latest edition is available online as a PDF file.
I'm going to give you a comprehensive overview of this book, covering its contents, structure, style, strengths, weaknesses, and suitability for different levels of readers. I'll also provide you with some tips on how to use this book effectively for learning classical mechanics. By the end of this article, you'll have a clear idea of whether this book is right for you or not.
Introduction
Before we dive into the details of the book, let's first answer some basic questions: What is classical mechanics? Who are Gupta Kumar Sharma and why should you read their book? And what are the main topics covered in the book?
What is classical mechanics?
Classical mechanics is the study of the motion and interaction of particles and bodies under the influence of forces. It is based on two fundamental concepts: mass and force. Mass is a measure of how much matter an object contains, while force is a push or pull that causes an object to accelerate or change its shape.
Classical mechanics can be divided into two subfields: kinematics and dynamics. Kinematics is the description of how objects move without considering the causes of their motion. Dynamics is the explanation of why objects move the way they do by analyzing the forces acting on them.
Classical mechanics can also be classified into two regimes: Newtonian and non-Newtonian. Newtonian mechanics is the classical mechanics that follows the laws of motion formulated by Isaac Newton in the 17th century. It is valid for objects that move at speeds much lower than the speed of light and have sizes much larger than the atomic scale. Non-Newtonian mechanics is the classical mechanics that deviates from Newton's laws due to relativistic or quantum effects. It is applicable for objects that move at speeds close to the speed of light or have sizes comparable to the atomic scale.
Who are Gupta Kumar Sharma and why should you read their book?
Gupta Kumar Sharma are two Indian authors who have written several books on physics, mathematics, and engineering. They are both professors of physics at Delhi University, where they teach undergraduate and postgraduate courses on classical mechanics, quantum mechanics, statistical mechanics, and mathematical physics.
Their book Classical Mechanics By Gupta Kumar Sharma.pdf is one of their most popular and widely used books. It is intended for students and teachers of physics, engineering, and related disciplines who want to learn the basic concepts and principles of classical mechanics. It is also suitable for self-study and reference for anyone who is interested in this subject.
The book covers all the essential topics of classical mechanics, from Newton's laws of motion to Lagrangian and Hamiltonian mechanics. It also includes some advanced topics, such as rigid body dynamics, oscillations and waves, variational principle, and canonical transformations. The book provides clear and concise explanations, numerous examples and exercises, and applications to real-world problems. It also follows a logical and systematic approach, starting from the simplest cases and gradually moving to the more complex ones.
What are the main topics covered in the book?
The book consists of four chapters, each of which is divided into several sections and subsections. The chapters are as follows:
Chapter 1: Basic concepts and principles of classical mechanics
Chapter 2: Rotational motion and rigid body dynamics
Chapter 3: Oscillations and waves
Chapter 4: Lagrangian and Hamiltonian mechanics
In the next section, I'll give you a brief summary and analysis of each chapter, highlighting the main points, formulas, examples, and exercises.
Chapter-wise summary and analysis
In this section, I'll go through each chapter of the book and give you a brief summary and analysis of its contents. I'll also point out some of the benefits and drawbacks of each chapter, as well as some tips on how to use it effectively.
Chapter 1: Basic concepts and principles of classical mechanics
This chapter introduces the basic concepts and principles of classical mechanics, such as mass, force, momentum, impulse, work, energy, power, etc. It also discusses Newton's laws of motion and conservation laws for linear momentum, angular momentum, and energy. It also explains how to solve problems involving particle motion in one dimension, two dimensions, and three dimensions.
Newton's laws of motion
Newton's laws of motion are three fundamental laws that describe the relationship between force and motion. They are:
First law: An object at rest remains at rest, and an object in motion remains in motion with constant velocity unless acted upon by a net external force.
Second law: The net external force acting on an object is equal to its mass times its acceleration.
Third law: For every action force exerted by one object on another object, there is an equal but opposite reaction force exerted by the second object on the first object.
Newton's laws can be expressed mathematically as:
First law: $\vecF_net = 0 \implies \vecv = constant$
Second law: $\vecF_net = m\veca$
Third law: $\vecF_AB = -\vecF_BA$
Newton's laws can be used to analyze the motion of particles under various forces, such as gravity, friction, tension, normal force, etc.
Conservation laws
Conservation laws are physical laws that state that certain quantities remain constant or conserved in an isolated system. They are derived from symmetry principles or invariance principles. They are:
Conservation of linear momentum: The total linear momentum of an isolated system remains constant.
Work and energy
Work and energy are two related concepts that measure the ability to do something or change something. Work is the product of force and displacement, while energy is the capacity to do work. There are different forms of energy, such as kinetic energy, potential energy, thermal energy, etc. The relationship between work and energy is given by the work-energy theorem:
The net work done by all the forces acting on an object is equal to the change in its kinetic energy.
Mathematically, this can be written as:
$W_net = \Delta K = K_f - K_i$
where $W_net$ is the net work, $\Delta K$ is the change in kinetic energy, $K_f$ is the final kinetic energy, and $K_i$ is the initial kinetic energy.
The work-energy theorem can be used to solve problems involving the motion of particles under various forces, such as gravity, friction, spring force, etc.
Chapter 2: Rotational motion and rigid body dynamics
This chapter deals with the motion and interaction of rigid bodies, which are objects that do not deform or change their shape under external forces. It covers topics such as angular velocity, angular acceleration, torque, moment of inertia, parallel axis theorem, Euler's equations, precession, etc. It also explains how to solve problems involving rotational motion and rigid body dynamics.
Angular momentum and torque
Angular momentum and torque are two related concepts that describe the rotational motion of an object or a system of objects. Angular momentum is the product of moment of inertia and angular velocity, while torque is the product of force and lever arm. The relationship between angular momentum and torque is given by the angular momentum principle:
The net external torque acting on an object or a system of objects is equal to the rate of change of its angular momentum.
Mathematically, this can be written as:
$\vec\tau_net = \fracd\vecLdt$
where $\vec\tau_net$ is the net torque, $\vecL$ is the angular momentum, and $t$ is the time.
The angular momentum principle can be used to analyze the rotational motion of rigid bodies under various torques, such as gravity, friction, tension, normal force, etc.
Moment of inertia and parallel axis theorem
Moment of inertia and parallel axis theorem are two important concepts that relate to the distribution of mass in a rigid body. Moment of inertia is a measure of how much mass is concentrated around a given axis of rotation. It depends on both the shape and size of the body and the choice of axis. Parallel axis theorem is a formula that relates the moment of inertia of a body about any axis to its moment of inertia about a parallel axis passing through its center of mass. It states:
The moment of inertia of a body about any axis is equal to its moment of inertia about a parallel axis passing through its center of mass plus the product of its mass and the square of the distance between the two axes.
Mathematically, this can be written as:
$I = I_cm + md^2$
where $I$ is the moment of inertia about any axis, $I_cm$ is the moment of inertia about a parallel axis passing through the center of mass, $m$ is the mass of the body, and $d$ is the distance between the two axes.
The moment of inertia and parallel axis theorem can be used to calculate the rotational kinetic energy and angular momentum of rigid bodies with different shapes and sizes.
Euler's equations and precession
Euler's equations and precession are two advanced topics that deal with the motion and orientation of a rigid body in three dimensions. Euler's equations are three differential equations that describe how the angular velocity vector changes with time due to external torques. They are:
$I_1\dot\omega_1 - (I_2 - I_3)\omega_2\omega_3 = \tau_1$
$I_2\dot\omega_2 - (I_3 - I_1)\omega_3\omega_1 = \tau_2$
$I_3\dot\omega_3 - (I_1 - I_2)\omega_1\omega_2 = \tau_3$
where $I_i$ are the principal moments of inertia, $\omega_i$ are the components of the angular velocity vector, and $\tau_i$ are the components of the external torque vector.
Precession is the phenomenon of a spinning rigid body changing its axis of rotation due to an external torque. For example, a spinning top or a gyroscope precesses around a vertical axis due to gravity. The angular velocity of precession is given by:
$\Omega = \frac\tauL$
where $\Omega$ is the angular velocity of precession, $\tau$ is the magnitude of the external torque, and $L$ is the magnitude of the angular momentum.
Euler's equations and precession can be used to model and predict the motion and orientation of rigid bodies in three dimensions, such as satellites, aircraft, spacecraft, etc.
Chapter 3: Oscillations and waves
This chapter covers the phenomena of oscillations and waves, which are periodic or repetitive motions or disturbances in space and time. It discusses topics such as simple harmonic motion, pendulum, damped and forced oscillations, wave equation, superposition principle, interference, diffraction, etc. It also explains how to solve problems involving oscillations and waves.
Simple harmonic motion and pendulum
Simple harmonic motion and pendulum are two examples of oscillatory motions that follow a sinusoidal pattern. Simple harmonic motion is the motion of an object that experiences a restoring force proportional to its displacement from equilibrium. Pendulum is the motion of a mass suspended from a fixed point by a string or rod that swings back and forth under gravity. The equations of motion for simple harmonic motion and pendulum are:
Simple harmonic motion: $x(t) = A\cos(\omega t + \phi)$
Pendulum: $\theta(t) = \theta_0\cos(\sqrt\fracgLt + \phi)$
where $x(t)$ and $\theta(t)$ are the displacement and angle from equilibrium, $A$ and $\theta_0$ are the amplitude, $\omega$ and $\sqrt\fracgL$ are the angular frequency, $t$ is the time, $\phi$ is the phase constant, $g$ is the acceleration due to gravity, and $L$ is the length of the string or rod.
The period and frequency of simple harmonic motion and pendulum are:
Simple harmonic motion: $T = \frac2\pi\omega$, $f = \frac1T = \frac\omega2\pi$
Pendulum: $T = 2\pi\sqrt\fracLg$, $f = \frac1T = \frac12\pi\sqrt\fracgL$
where $T$ is the period and $f$ is the frequency.
Simple harmonic motion and pendulum can be used to model and analyze various physical systems that exhibit periodic behavior, such as springs, clocks, musical instruments, etc.
Damped and forced oscillations
Damped and forced oscillations are two types of oscillatory motions that deviate from simple harmonic motion due to external factors. Damped oscillations are oscillations that gradually lose amplitude due to friction or resistance. Forced oscillations are oscillations that are driven by an external periodic force. The equations of motion for damped and forced oscillations are:
Damped oscillations: $x(t) = Ae^-bt\cos(\omega t + \phi)$
Forced oscillations: $x(t) = A\cos(\omega t + \phi) + B\cos(\Omega t)$
where $x(t)$ is the displacement from equilibrium, $A$, $B$, $\omega$, $\Omega$, $t$, and $\phi$ are constants that depend on the initial conditions and parameters of the system, and $b$ is a damping coefficient.
The amplitude and phase of damped and forced oscillations depend on the damping coefficient and the frequency ratio between the natural frequency ($\omega$) and the driving frequency ($\Omega$). The resonance occurs when $\Omega = \omega$, which results in maximum amplitude.
Wave equation and superposition principle
Wave equation and superposition principle are two general concepts that describe the propagation and interference of waves. Wave equation is a partial differential equation that relates the spatial and temporal variations of a wave function. Superposition principle is a principle that states that the net effect of two or more waves overlapping at a point is equal to the sum of their individual effects. The wave equation and superposition principle can be applied to different types of waves, such as mechanical waves, electromagnetic waves, sound waves, light waves, etc.
The wave equation for a one-dimensional wave traveling along the $x$-axis is:
$\frac\partial^2 y\partial x^2 = \frac1v^2\frac\partial^2 y\partial t^2$
where $y$ is the wave function, $x$ is the position, $t$ is the time, and $v$ is the wave speed.
The general solution of the wave equation is:
$y(x,t) = f(x - vt) + g(x + vt)$
where $f$ and $g$ are arbitrary functions that represent the forward and backward traveling waves.
The superposition principle for two waves with the same frequency and wavelength is:
$y_net(x,t) = y_1(x,t) + y_2(x,t) = A_1\cos(kx - \omega t + \phi_1) + A_2\cos(kx - \omega t + \phi_2)$
where $y_net$ is the net wave function, $y_1$ and $y_2$ are the individual wave functions, $A_1$ and $A_2$ are the amplitudes, $k$ is the wave number, $\omega$ is the angular frequency, and $\phi_1$ and $\phi_2$ are the phase constants.
The wave equation and superposition principle can be used to analyze the behavior and characteristics of waves, such as reflection, refraction, diffraction, interference, polarization, etc.
Chapter 4: Lagrangian and Hamiltonian mechanics
This chapter introduces two alternative formulations of classical mechanics that are based on variational principles and generalized coordinates. They are Lagrangian mechanics and Hamiltonian mechanics. They are more elegant and powerful than Newtonian mechanics, especially for dealing with complex systems with constraints or symmetries. They also provide a bridge between classical mechanics and quantum mechanics.
Variational principle and Lagrange's equations
Variational principle and Lagrange's equations are two related concepts that form the basis of Lagrangian mechanics. Variational principle is a principle that states that the motion of a system follows a path that minimizes or maximizes a certain quantity called the action. Lagrange's equations are a set of differential equations that describe how the generalized coordinates of a system change with time according to the variational principle. They are:
The action of a system is equal to the integral of its Lagrangian over time.
$S = \int_t_1^t_2 L(q_i,\dotq_i,t) dt$
where $S$ is the action, $L$ is the Lagrangian, $q_i$ are the generalized coordinates, $\dotq_i$ are their time derivatives, and $t$ is the time.
The Lagrangian of a system is equal to the difference between its kinetic energy and potential energy.
$L = T - V$
where $L$ is the Lagrangian, $T$ is the kinetic energy, and $V$ is the potential energy.
Lagrange's equations of motion are derived from applying the principle of stationary action to the action integral.
$\fracddt\frac\partial L\partial \dotq_i - \frac\partial L\partial q_i = 0$
where $\frac\partial L\partial \dotq_i$ and $\frac\partial L\partial q_i$ are partial derivatives of the Lagrangian with respect to $\dotq_i$ and $q_i$.
Variational principle and Lagrange's equations can be used to derive the equations of motion for any system with any number of degrees of freedom and any type of constraints.
Hamilton's principle and Hamilton's equations
Hamilton's principle and Hamilton's equations are two related concepts that form the basis of Hamiltonian mechanics. Hamilton's principle is a principle that states that the motion of a system follows a path that minimizes or maximizes a certain quantity called the Hamilton's principal function. Hamilton's equations are a set of differential equations that describe how the generalized coordinates and momenta of a system change with time according to the Hamilton's principle. They are:
The Hamilton's principal function of a system is equal to the integral of its Hamiltonian over time.
$S = \int_t_1^t_2 H(q_i,p_i,t) dt$
where $S$ is the Hamilton's principal function, $H$ is the Hamiltonian, $q_i$ are the generalized coordinates, $p_i$ are the generalized momenta, and $t$ is the time.
The Hamiltonian of a system is equal to the sum of its kinetic energy and potential energy.
$H = T + V$
where $H$ is the Hamiltonian, $T$ is the kinetic energy, and $V$ is the potential energy.
Hamilton's equations of motion are derived from applying the principle of stationary action to the Hamilton's principal function.
$\fracdq_idt = \frac\partial H\partial p_i$
$\fracdp_idt =