# Mechanics

Introduction

Mechanics is the branch of physics that deals with the motion of objects and the forces that act on them. It is the foundation of many other branches of physics, such as thermodynamics, electromagnetism, and quantum mechanics.

Vectors

Vectors are quantities that have both magnitude and direction. They are used to represent many physical quantities, such as force, velocity, and acceleration.

Motion in One Dimension

One-dimensional motion is motion along a straight line. Some of the important concepts in one-dimensional motion include:

·         Velocity: The rate of change of position with respect to time.

·         Acceleration: The rate of change of velocity with respect to time.

·         Free fall: The motion of an object under the influence of gravity alone.

Projectile Motion

Projectile motion is the motion of an object that is thrown or projected into the air. It is a special case of two-dimensional motion.

Newton's Laws of Motion

Newton's laws of motion are the three fundamental laws that describe the motion of objects. They are:-

·         First law: An object at rest will remain at rest, and an object in motion will remain in motion at a constant velocity, unless acted upon by an external force.

·         Second law: The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

·         Third law: For every action, there is an equal and opposite reaction.

Work and Energy

Work is the transfer of energy from one system to another. Energy is the ability to do work. There are two main types of energy: kinetic energy and potential energy.

·         Kinetic energy: The energy of a moving object.

·         Potential energy: The energy stored in an object due to its position or configuration.

Momentum

Momentum is a measure of the inertia of an object. It is defined as the product of the object's mass and velocity.

Conservation of Momentum

In a closed system, the total momentum is always conserved. This means that the momentum of an object can only change if it interacts with another object.

Rotational Motion

Rotational motion is the motion of an object around an axis. Some of the important concepts in rotational motion include:

·         Angular velocity: The rate of change of angular position with respect to time.

·         Angular acceleration: The rate of change of angular velocity with respect to time.

·         Torque: The rotational equivalent of force.

·         Angular momentum: The rotational equivalent of momentum.

·         Conservation of angular momentum: In a closed system, the total angular momentum is always conserved.

Mechanics

Mechanics is the branch of Physics dealing with the study of motion when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.

There are two sub-disciplines Classical Fields and Quantum Fields

Classical mechanics

Classical mechanics deals with the question of how an object moves when it is subjected to various forces, and also with the question of what forces act on an object which is not moving.

The word “classical” indicates that we are not discussing situations in which an object moves with a velocity which is an appreciable fraction of the velocity of light or phenomena on the atomic scale. The description of atomic phenomena requires quantum mechanics, and the description of phenomena at very high velocities requires Einstein’s Theory of Relativity. Both quantum mechanics and relativity were invented in the twentieth century; the laws of classical mechanics were stated by Sir Isaac Newton in 1687. The laws of classical mechanics enable us to calculate the trajectories of bullets and baseballs, planets and space vehicles. Using these laws we can predict the position-versus-time relation for a cylinder rolling down an inclined boat or for an oscillating pendulum and can calculate the tension in the wire when a picture is hanging on a wall.

The practical importance of the subject hardly requires demonstration in a world which contains buildings, automobiles, aeroplanes, ballistic missiles, and bridges, for a person who does not have any professional reason there is a compelling intellectual reason to study classical mechanics: this is the example par excellence of a theory which explains an incredible multitude of phenomena on the basis of a minimal number of simple principles.

Quantum Mechanics

Quantum mechanics can be thought of roughly as the study of physics on very small length scales, although there are also certain macroscopic systems it directly applies to. The descriptor “quantum” arises because, in contrast with classical mechanics, certain quantities take on only discrete values. However, some quantities still take on continuous values, In quantum mechanics, particles have wavelike properties, and a particular wave equation, the Schrodinger equation, governs how these waves behave. The Schrodinger equation is different in a few ways from the other wave equations we’ve seen in this book. But these differences won’t keep us from applying all of our usual strategies for solving a wave equation and dealing with the resulting solutions.

In some respect, quantum mechanics is just another example of a system governed by a wave equation. However, although it is fairly straightforward to deal with the actual waves, there are many things about quantum mechanics that are a combination of subtle, perplexing, and bizarre. To name a few: the measurement problem, hidden variables along with Bell’s theorem, and wave-particle duality. Even though there are many things that are highly confusing about quantum mechanics, the nice thing is that it’s relatively easy to apply quantum mechanics to a physical system to figure out how it behaves.

Statistical Mechanics

This branch of statistical mechanics which treats and extends classical thermodynamics is known as statistical thermodynamics or equilibrium statistical mechanics.

We apply statistical mechanics to solve real systems (a system for many particles). We can easily solve Schrodinger’s equation for 1 particle, atom, or molecule. For many particles, the solution will take the form:

Total = linear combination of a (1) b (2) c (3)…

Where a means a particle in the state with an energy Ea.

For example in an ideal gas, We assume that the molecules are non-interacting, i.e. they do not affect each other’s energy levels. Each particle contains a certain energy.

At T > 0, the system possesses total energy, E.

So, how is E distributed among the particles? Another question would be, how the particles are distributed over the energy levels.

We apply statistical mechanics to provide the answer and thermodynamics demands that the entropy is maximum at equilibrium.

Thermodynamics is concerned with heat and the direction of heat flow, whereas statistical mechanics gives a microscopic perspective of heat in terms of the structure of matter and provides a way of evaluating the thermal properties of matter, for e.g., heat Capacity.