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\begin{document}

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\begin{center}
\large{\textbf{Indian Institute of Information Technology Allahabad}}
\end{center}
\vspace{-25pt}
\begin{center}
\large{\textbf{Review Test- Classical Mechanics}}
\end{center}
\hfill \footnotesize{Date: 04/09/2018}
\vspace{-18pt}

\noindent
\hrulefill
\vspace{-17pt}
\begin{center}
\footnotesize{B.Tech. IT, ECE - Semester I}
\end{center}
\vspace{-20pt}

\begin{center}
\footnotesize{Paper Code: SEGP132C}
\end{center}
\vspace{-20pt}

%\begin{center}
%\footnotesize{Paper Setter: Abdullah Bin Abu Baker \& Sumit Kumar Upadhyay}
%\end{center}
\vspace{-10pt}
\noindent
\footnotesize{Max Marks: 20} \hfill \small{Duration: 1 hour}

\noindent
\footnotesize{Attempt \textbf{all} the questions. There is \textbf{no credit} for an answer if proper justification is not given, even if the answer is correct. Notations are standard. \textbf{Do not write} on question paper and cover pages except your details.}

\noindent
\hrule

\singlespacing
\thispagestyle{empty}
\begin{enumerate}

 \item A freely moving particle of mass $m$ is moving with velocity $\bf{v}=\dot{r}$ with respect to an inertial reference frame $K$. Let $K'$ be another inertial frame moving with a constant velocity $\bf{V}$ w.r.t. frame $K$. Write down the Lagrangian w.r.t. the frame $K'$ and show that the Euler-Lagrange equation remains the same.  \hfill[3]
  \item A mass $m$ is suspended from the ceiling by a string of length $l$. The mass is free to move in all directions. This arrangement is called spherical pendulum. Write down the equation of constraint in terms of cartesian coordinates for this system. Is this constraint holonomic?  Obtain the Lagrangian and find out equations of motion for the mass. Identify the cyclic coordinate and corresponding conserved quantity. Find out generalized momenta for the system. Obtain the Hamilton's equations of motion. \hfill[1+1+1+1+2+1+1+2+2+2]
 \item Using Hamilton's principle directly show that equation of motion for a $1$ dimensional simple harmonic oscillator of mass $m$ is \[\ddot{x}=- {k\over m} x,\] where $k$ is Force constant.\hfill[3]
 

    
\end{enumerate}
\vspace{2.5cm}
\parskip = 8pt

\begin{center}
\large{\textbf{Indian Institute of Information Technology Allahabad}}
\end{center}
\vspace{-25pt}
\begin{center}
\large{\textbf{Review Test- Classical Mechanics}}
\end{center}
\hfill \footnotesize{Date: 04/09/2018}
\vspace{-18pt}

\noindent
\hrulefill
\vspace{-17pt}
\begin{center}
\footnotesize{B.Tech. IT, ECE - Semester I}
\end{center}
\vspace{-20pt}

\begin{center}
\footnotesize{Paper Code: SEGP132C}
\end{center}
\vspace{-20pt}

%\b
%\footnotesize{Paper Setter: Abdullah Bin Abu Baker \& Sumit Kumar Upadhyay}
%\end{center}
\vspace{-10pt}
\noindent
\footnotesize{Max Marks: 20} \hfill \small{Duration: 1 hour}

\noindent
\footnotesize{Attempt \textbf{all} the questions. There is \textbf{no credit} for an answer if proper justification is not given, even if the answer is correct. Notations are standard. \textbf{Do not write} on question paper and cover pages except your details.}

\noindent
\hrule

\singlespacing
\thispagestyle{empty}
\begin{enumerate}

 \item A freely moving particle of mass $m$ is moving with velocity $\bf{v}=\dot{r}$ with respect to an inertial reference frame $K$. Let $K'$ be another inertial frame moving with a constant velocity $\bf{V}$ w.r.t. frame $K$. Write down the Lagrangian w.r.t. the frame $K'$ and show that the Euler-Lagrange equation remains the same.  
  \hfill[3]
  \item A mass $m$ is suspended from the ceiling by a string of length $l$. The mass is free to move in all directions. This arrangement is called spherical pendulum. Write down the equation of constraint in terms of cartesian coordinates for this system. Is this constraint holonomic?  Obtain the Lagrangian and find out equations of motion for the mass. Identify the cyclic coordinate and corresponding conserved quantity. Find out generalized momenta for the system. Obtain the Hamilton's equations of motion. \hfill[1+1+1+1+2+1+1+2+2+2]
 \item Using Hamilton's principle directly show that equation of motion for a $1$ dimensional simple harmonic oscillator of mass $m$ is \[\ddot{x}=- {k\over m} x,\] where $k$ is Force constant.   \hfill[3]
 

    
\end{enumerate}
\vspace{3.5cm}
\end{document}