Classical Mechanics and Mathematical Methods in Physics

MSc Physics 1st Year

Q.1 Write short notes on the following

1. Law of inertia

Ans: Law of inertia: Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.

The First Law of Motion, commonly called the “Law of Inertia,”

Explanation: When a body moves with constant velocity, there are either no forces present or there are forces acting in opposite directions that cancel out. If the body changes its velocity, then there must be an acceleration, and hence a total non-zero force must be present. We note that velocity can change in two ways. The first way is to change the magnitude of the velocity; the second way is to change its direction.

For example: a package thrown out of an airplane will continue to move at the speed of the airplane on the horizontal axis (in the direction of the airplane's movement). Since the law of gravity acts on the package (a vertical downward axis), the package will gather speed along the vertical axis, but on the horizontal axis its speed will remain equal to that of the airplane.

2. Constrained motionAns: Constrained motion:

Whenever links are assembled to synthesize a mechanism, certain restrictions on free movement get associated with each link. This restrictions are referred to as constraints and any relative motion

arising thereof is known as a constrained motion

Basically there are three types of constrained motion

1. Completely Constrained Motion : When a motion between the pair takes place in a definite direction w.r.t. direction of the force applied then the motion is called Completely Constrained Motion

Square bar in a square hole undergoes completely constrained motion

2. Successfully constrained motion. If constrained motion is not achieved by the pairing

elements themselves, but by some other means, then, it is called successfully constrained

motion. Eg. Foot step bearing, where shaft is constrained from moving upwards, by its self

weight.

3. Incompletely constrained motion . When relative motion between pairing elements takes

place in more than one direction, it is called incompletely constrained motion. Eg. Shaft in a

circular hole.

3. Frame of reference

Ans: Frame of reference:

A set of coordinate axis in terms of which position or movement may be specified or with reference to which physical laws may be mathematically stated also called frame of reference

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Classical Mechanics and Mathematical Methods in Physics

MSc Physics 1st Year

A frame of reference is a reference point combined with a set of directions. A graphical representation of a 1-dimensional frame of reference:

For example, a boy is standing still inside a train as it pulls out of a station. Both you and the boy define your location as the point of reference and the direction train is moving as a where you are standing as the point of reference and the direction the train is moving in as forward

4. Galilean and Lorentz transformation.

Ans: Galilean and Lorentz transformation.

Lorentz transformation:

The primed frame moves with velocity v in the x direction with respect to the fixed reference frame. The reference frames coincide at t=t'=0. The point x' is moving with the primed frame.

The reverse transformation is:

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Classical Mechanics and Mathematical Methods in Physics

MSc Physics 1st Year

Much of the literature of relativity uses the symbols and as defined here to simplifyβ γ the writing of relativistic relationships.

5. Michelson-Morley’s experimentAns: Michelson-Morley’s experiment: The most famous and successful was the one now known as the Michelson-Morley experiment, performed by Albert Michelson   (1852-1931) and Edward Morley   (1838-1923) in 1887.

The Michelson Morley experiment is not consistent with Galilean/Newtonian relativity, as the introductory film clip shows. Its results are explained using Einstein's principle of relativity. A non-quantitative introduction is given here. This page gives a simple quantitative analysis.

Schematically, a beam of monochromatic light is divided by a beam splitter (a transparent sheet at an angle). The divided beams reach two mirrors, are returned and recombined by respectively transmission and reflection at the beam splitter. Their relative phase produces an interference pattern in the combined beam.

Schematic of the Michelson-Morley experiment. A beam of light (actually continuous, not pulsed as in the animation below) is split when it strikes a transparent plate: part is transmitted, part reflected. When the two divided beams return to the block, partial reflection and partial transmission also combines them.

Following Galilean/Newtonian physics, let us suppose that light travels (at c) with respect to a 'stationary' medium (called the æther). For our purposes, suppose that it be set up with l1 = l2 and that the whole spectrometer be stationary with respect to the æther, the medium that supports the wave motion of light. Let's consider a point in the interference pattern at which the phase differene is zero. This is the situation shown at left.

Now suppose that it move to the right at speed v with respect to æther. Picture this from a frame at rest in the æther (middle diagram). The transit times are no longer equal, but for the horizontal and near vertical directions are given by:

By rotating the spectrometer 90 degrees, one can compare the effect of speed through the putative æther on one of the beams. Then by making measurements six months apart, one can add or subtract the speed of the Earth through æther. The speed of the Earth in its orbit around the sun is v = 30 km/s. Substituting in the equations above (and using l = 11 m - for an optical spectrometer, it was a seriously large!) the phase difference expected would be

= 2 t(c/ ) = 2.3 radians = 0.4 fringes.Δφ πΔ λ

The spectrometer was easily sensitive enough to see this*. However, the result was: 0.00 plus or minus 0.01 fringes.

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Classical Mechanics and Mathematical Methods in Physics

MSc Physics 1st Year

Q.2. Explain angular momentum for a system of particles.Ans: Angular momentum for a system of particles: Angular momentum is a vector quantity (more precisely, a pseudo vector) that represents the product of a body's rotational inertia and rotational velocity about a particular axis

let a system of particles is made up of n number of particles. Let ri be the position vector of the ith particle P with respect to a point O and vi be its velocity .let Rcm be the position vector of center of mass C of the system with respect to the origin

Let ri' and vi

' be the position vector and velocity vector of the ith particle with respect to center of mass of the system.

Angular momentum of the system of particles with respect to origin is given by

Angular momentum of the system of particles with respect to center of mass of the system is given by

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Classical Mechanics and Mathematical Methods in Physics

MSc Physics 1st Year

Hence the angular momentum of the system of the particles with respect to point O is equal to the sum of the angular momentum of the center of mass of the particles about O and angular monentum of the system about center of mass

Law of conservation of angular momentum Torque acting on any particle is given by

 If external torque acting on any particle os zero then,

  Hence in absence of external torque the angular momentum of the particle remains constant or

conserved. 

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Classical Mechanics and Mathematical Methods in Physics

MSc Physics 1st Year

Total torque acting on any system is given by

  If total external force acting on any particle system is zero or,

  If total external torque acting on any body is zero, then total angular momentum of the body remains

constant or conserved.

Q.3.Differentiate general and special theory of relativity.

Ans: Difference Between General Theory of Relativity and Special Theory of Relativity.

General Theory of Relativity Special Theory of Relativity1. General Relativity is the theory that

that aforementioned four dimensional manifold has curvature, and that curvature is induced by the presence of mass/energy in space-time

Special relativity is the theory that space-time is a four-dimensional manifold with metric signature (+,-,-,-). It explains the speed of light having a maximum, time dilation, relativistic effects, and energy-mass-equivalence.

2. General relativity is called such because it can be applied generally. 

Special relativity is called special simply because it is a special case of general relativity. 

3. General relativity deals with everything else - inertial frames of reference WRT low velocities, and also with mass interactions involved.

Special relativity deals with the rather specific application of inertial frames of reference WRT very high velocities.

4. General relativity deals with [STRIKE] accelerating reference frames and [/STRIKE] curved space....

Special relativity deals with [STRIKE] inertial reference frames and[/STRIKE] flat space. 

5. General relativity applies to all coordinate systems.

Special relativity applies only to coordinate systems which correspond to inertial frames 

Q.5. What do you understand bye Hamilton’s variation principle? Explain it. 

Ans: Hamilton’s variation principle: It states that the dynamics of a physical system is determined by

a variation problem for a functional based on a single function, the Lagrangian, which contains all physical

information concerning the system and the forces acting on it. The variation problem is equivalent to and

allows for the derivation of the differential equations of motion of the physical system. Although formulated

originally for classical mechanics, Hamilton's principle also applies to classical fields such as the

electromagnetic and gravitational fields, and has even been extended to mechanics, quantum and criticality

theories.

Explanation of Hamilton’s variation principle:

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Classical Mechanics and Mathematical Methods in Physics

MSc Physics 1st Year

Illustration of the variation in a generalized q between times t1and t2.

Hamilton's principle states that the true evolution q(t) of a system described by N 

Generalized coordinates q = (q1, q2, ..., qN) between two specified states q1 = q(t1) and q2 = q(t2) at two specified times t1 and t2 is a stationary point

 (A point where the variation is zero), of the action functional

where   is the Lagrangian function for the system.

In other words, any first-order perturbation of the true evolution results in (at most) second-

order changes in  . The action   is a functional,

i.e., something that takes as its input a function and returns a single number, a scalar. In terms

of functional analysis, Hamilton's principle states that the true evolution of a physical system is a solution of the functional equation

Hamilton's principle

Hamilton's principle is an important variation principle in electrodynamics. As opposed to a system composed of rigid bodies, deformable bodies have an infinite number of degrees of freedom and occupy continuous regions of space; consequently, the state of the system is described by using continuous functions of space and time. The extended Hamilton Principle for such bodies is given by

where T is the kinetic energy, U is the elastic energy, We is the work done by external loads on the body, and t1, t2 the initial and final times. If the system is conservative, the work done by external forces may be derived from a scalar potential V. In this case,

This is called Hamilton's principle and it is invariant under coordinate transformations.

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