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### CM-03 :: Action Principle

Summary (or abstract) not available for this node.
kapoor 6 days 21 hours ago

### Problem/CM-03001

[A] Calculate the value of the action integral between the limits $t=0$
and $t=T$ for a a particle falling under influence of gravity along the
following three paths.

(i)  for a fictitious motion with path given by $z= at.$
(ii) for a second fictitious path given by $z=bt^3.$
(iii) for the real motion $z={1\over 2} g t^2.$

where the constants $a, b$ must be determined so that the initial and
final positions coincide with the rules of variation in the action
principle.

[B] Check if  the action integral has smaller value for the real motion
in(c) than the fictitious ones (a) and (b). Discuss the result you have
obtained and write conclusions you may draw about the action principle.

kapoor 6 days 21 hours ago

### QM-02 Vetor Spaces

Summary (or abstract) not available for this node.
kapoor 6 days 21 hours ago

### Problem/CM-09001

Show that the kinetic energy of a uniform rod of mass $$m$$ is
$T = \frac{1}{6}m(\vec{u}.\vec{u} + \vec{u}.\vec{v}+\vec{v}.\vec{v} )$
where $$\vec{u}$$ and $$\vec{v}$$ are the velocities of the two ends.

Woodhouse

kapoor 6 days 21 hours ago

### Problem/CM-02023 Generalized potential for a charged particle in EM fields

$$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$$

A point charge moves with velocity $$\vec{v}$$ in presence of electric field $$\vec{E}$$ and magentic field $$\vec{B}$$. The Lorentz force on the charged particle is
$\vec{F}= q\Big\{ \vec{E} + \frac{1}{c}\vec{v}\times\vec{B} \Big\}$
In terms of the scalar potential $$\phi$$ and vector  potential $$\vec{A}$$ related to the electric and magnetic fields by
$\vec{E}= -\nabla \phi- \frac{1}{c} \pp[\vec{A}]{t}, \quad \vec{B}=\nabla\times \vec{A}.$
the Lorentz force becomes $\vec{F} = q\Big\{ - \nabla\phi -\frac{1}{c} \pp[\vec{A}]{t} + \frac{1}{c} \vec{v}\times(\nabla \times \vec{A})\Big\}$
Determine the generalzed potential for this system and write the Lagrangian.

kapoor 6 days 21 hours ago

### Problem/TH-04002

The molar energy of a monoatomic gas which obeys van der Waal's equation is given by
$$E= \frac{3}{2}kT - \frac{a}{v}$$,
where $$V$$ is the volume at temperature $$T$$, and $$a$$ is a constant. Initially one mole of gas is at temperature $$T_1$$ and occupies volume $$V_1$$. The gas is allowed to  expand adiabatically into a vacuum so that it occpies a total volume $$V_2$$. What is the final temperature of the gas?

MANDL

kapoor 6 days 21 hours ago

### Question/EM-02001 Recalling and Understanding -- Short Questions

Summary (or abstract) not available for this node.
kapoor 6 days 21 hours ago

### Question/EM-02002 Analyzing and Applying Problems

Summary (or abstract) not available for this node.
kapoor 6 days 21 hours ago

### CM-08 :: Motion in Non-Inertial Frames

Summary (or abstract) not available for this node.
kapoor 6 days 21 hours ago

### CM-07 S:: mall Oscillations

Summary (or abstract) not available for this node.
kapoor 6 days 21 hours ago