Advanced Classical Mechanics
Prof. Iain W. Stewart
Classical Mechanics III
MIT 8.09 & 8.309
Editors: Francisco Machado, Thanawuth Thanathibodee,
Prashanth S. Venkataram
c 2016 Iain W. Stewart
Contents
1 A Review of Analytical Mechanics
1.1 Introduction .
Indian Institute of Technology Guwahati
Department of Physics
PH201/Advanced Classical Mechanics/2017-18/Tutorial-1/AKSharma
Due date: 07.08.17
1.
A string of length l1 passes over a light fixed pulley, supporting a mass m1 on one end and a
pulley of mass
Indian Institute of Technology Guwahati
Department of Physics
PH201/Advanced Classical Mechanics/2017-18/Tutorial-2/AKSharma
Due date: 14.08.17
1. In general the integrand f(y, , x) whose integral we wish to minimize depends on y, , and x.
There is a cons
Indian Institute of Technology Guwahati
Department of Physics
PH201/Advanced Classical Mechanics/2017-18/Tutorial-3/AKSharma
Due date: 21.08.17
1. A string of length l1 passes over a light fixed pulley, supporting a mass m1 on one
end and a pulley of mass
2016\PH102\Tutorial 3
Physics II
1. A spherical surface of radius R and center at origin carries a surface charge (, ) = 0 cos .
Find the electric field at z on z-axis. Treat the case
z < R (inside) as well as z > R (outside).
[Hint: Be sure to take the
2016\PH102\Tutorial 6
Physics II
1. A point charge q is imbedded at the center of a sphere of linear dielectric material (with susceptibility e
~ Find the electric field, the polarization, and the bound charge densities, b and b . What
and radius R).
is t
2016\PH102\Tutorial 2
Physics II
+3y z and S is that part of the plane 2x+3y+6z =
1. Evaluate A
n ds, where A = 18z x12 y
12 which is located in the first octant.
R
R
and S is the surface of the cylinder x2 +y 2 = 16
2. Evaluate S A
n ds, where A = zi+x
2016\PH102\Tutorial 4
Physics II
1. [G 3.1] Find the average potential over a spherical surface of radius R due to a point charge q located
inside. Show that in general,
Qenc
Vave = Vcenter + 4
,
0R
where Vcenter is the potential at the center due to all
2016\PH102\Tutorial 10
Physics II
~ is approaching toward a circular loop of radius a along its axis with a
1. A small magnet of moment M
constant speed v. Show that when the magnet is at a distance x from the coil the current in the loop is
given by
30 M
2016\PH102\Tutorial 1
Physics II
Note:
The main purpose of the tutorial is to provide you with an opportunity to interact with a teacher.
The teacher will assist you in clearing your doubts and answer your queries regarding the topics covered
in lecture
2016\PH102\Tutorial 8
Physics II
1. Currents I and I flow along lines at y = d/2 and y = d/2 , respectively, on the yz plane, as shown in
Figure. Determine the vector potential at point P sufficiently far from the z-axis.
Figure 1: Problem 1
2. If B is un
2016\PH102\Tutorial 7
Physics II
~
1. An electron is injected with a velocity ~u0 = yb u0 at the origin into a region where both an electric field E
~
~
~
b B0 . Discuss the
and a magnetic field B exist. Describe the motion of the electron if E = zb E0 ,a
2016\PH102\Tutorial 5
Physics II
1. [G 4.2] According to quantum mechanics, the electron cloud for a hydrogen atom in the ground state has
a charge density
q 2r/a
(r) =
e
,
a3
where q is the charge of the electron and a is the Bohr radius. Find the atomic
Lecture-VIII
Work and Energy
The problem:
As first glance there seems to be no problem in finding the motion of a particle if we
know the force; starting with Newton's second law, we obtain the acceleration, and by
integrating we can find first the veloci
Lecture-XVIII
Rigid body motion
Equation of motion
In the body frame: L = I
dL
dL
= + L,
dt fix dt body
dL
= + L
dt body
Therefore,
If the body frame coincides with the principal axes of the system, the moment of
inertia tensor will be gi
Lecture-XIV
Noninertial systems
Accelerating frame
Newton's second law F = ma holds true only in inertial coordinate systems.
However, there are many noninertial (that is, accelerating) frames that one needs to
consider, such as elevators, merry-go-rounds
Lecture-IX
Work and Energy
Non-conservative forces:
Suppose both conservative and non-conservative forces are acting on the same
system. For instance, an object falling through the air experiences the conservative
gravitational force and the non-conservat
Lecture-XVII
Rigid body motion
Coordinate transformation:
y
y
(x,y,z); (x,y,z)
x = x cos + y sin
y = x sin + y cos
z = z
x cos
y = sin
z 0
sin
cos
0
x
0 x
x
0 y = R3 ( ) y
z
1 z
x
z,z
Rotation about z-axis:
cos sin
R3 ( ) = sin co
Lecture-XVI
Rigid body motion
General motion of a rigid body
Eulers theorem: Any general displacement of a rigid body, one point of which is fixed,
is a rotation about some axis passing through the fixed point.
Two coordinate systems will be used: one is
Bohr Atom and Specific Heats of Gases and Liquids
S Uma Sankar
Department of Physics
Indian Institute of Technology Bombay
Mumbai, India
S. Uma Sankar (IITB)
Lecture-3
1 August 2016
1 / 25
Kirschhoffs Law
Kirchhoff proved the following theorem based on th
Lecture-X
Small oscillation
Stability:
The result F = -dU/dx is useful not only for computing the force but also for visualizing
the stability of a system from the potential energy plot.
Suppose there is a force on the particle is F = -dU/dx, and the syst
Lecture-1
Polar coordinates
Polar coordinates:
r
z = zP
The coordinates (r, ) in the
plane z = zP are called polar
coordinates.
Polar and Cartesian coordinates:
If polar coordinates (r, ) of a point in the plane
are given, the Cartesian coordinates (x, y)
Lecture-XV
Noninertial systems
Apparent Force in Rotating Coordinates
The force in the rotating system is
where
The first term is called the Coriolis force, a velocity dependent force and the second
term, radially outward from the axis of rotation, is cal
The Quantum Harmonic Oscillator
Consider the familiar problem of a mass tied to a spring. In classical physics we
know that the position and momentum of the particle at time t is given by,
x(t) = A cos(t + )
and momentum is,
p(t) = mx(t)
= Am sin(t + )
is
The story so far .
Einstein and de Broglie together wrote down the mathematical
formulae that explained the meaning of wave particle duality
=
=
T=
Schrodinger used this to write down the expression for the
probability amplitude of a matter wave with
Wave Mechanics
A Mathematical Description
of Matter Waves
Schrodingers Equation
In January 1926, Schrdinger
published in Annalen der
Physik
the
paper
"Quantisierung
als
Eigenwertproblem"
[tr.
Quantization as an Eigenvalue
Problem] on wave mechanics
and pr
PH204: Quantum Mechanics (3-1-0-8)
Review of wave mechanics: Youngs double slit, de Broglie relation, wave packets, Schrodinger
equation; Observable, Eigen values and Eigen functions; Simple applications: particle in a box;
potential well; potential barri
HYDROGEN ATOM IN QUANTUM MECHANICS
An electron bound to the nucleus is described by an elliptical or circular trajectory in classical physics. In quantum
physics we know that there is no trajectory rather a probability amplitude that describes the likelih
Generator of translations and rotations
1
Generator of Translations
Consider a function (x). I want to find an operator that takes this and maps it to
(x + a). Thus I want to find Ta such that for any (x),
Ta (x) = (x + a)
Imagine I expand the right hand