\documentclass[10pt]{article}%\makeatletter
\oddsidemargin .0in \evensidemargin .0in \textwidth 6.5in
\topmargin -.30in \textheight 24cm
\usepackage{amsmath,amssymb,amsthm,mathrsfs,booktabs,graphics,graphicx,float,mathtools,color}
\usepackage[mathscr]{euscript}
\usepackage{setspace}
%\usepackage{enumitem}
\newcommand{\boldr}{\boldsymbol{r}}
\newcommand{\boldi}{\boldsymbol{i}}
\newcommand{\boldj}{\boldsymbol{j}}
\newcommand{\boldk}{\boldsymbol{k}}
\newcommand{\boldv}{\boldsymbol{v}}
\newcommand{\rt}[1]{\textcolor{red}{#1}}
%\newcommand{\rt}[1]{\textcolor{red}{#1}}
\newcommand{\rh}[1]{\hfill{\rt{{#1}}}}

% =============================================================== %
%      List environment with lower case alphabetic counter        %
% =============================================================== %
\newcounter{list-counter}
\newenvironment{listalph}
{\begin{list}{(\alph{list-counter})~}{\usecounter{list-counter}}
%\setlength{\itemsep}{-0.1in}
%\setlength{\topsep}{0in}
}
{\end{list}}

\begin{document}

\parskip = 10pt

\begin{center}
\large{\textbf{Indian Institute of Information Technology Allahabad}}
\end{center}
\vspace{-25pt}
\begin{center}
\large{\textbf{Quiz I}}
\end{center}
\hfill \footnotesize{Date: 19/09/2017}
\vspace{-18pt}

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

\begin{center}
\footnotesize{Paper Code: SPHY132C}
\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 Assume a frictionless, massless pulley of radius $R$ and a massless rope of length $l$. Two masses $m_1$ and $m_2$ are attached at the two ends of the rope. Let at an instant, the coordinates of the masses are $(-x_1, y_1)$ and $(x_1,y_2)$ respectively (with respect to a Cartesian coordinate system erected at the centre of the pulley). What constraint (equation) does this system of masses satisfy? What type of constraint (holonomic or non-holonomic) is this? Is the constraint sceleronomous or rheonomous? What is the number of degrees of freedom of this system? Compute the Lagrangian of the system. Hence deduce the Hamiltonian of the system. Find out the equation of motion. \hfill[1+1+1+1+3+2+1]




\item A double plane pendulum consists of two simple pendulums, with one pendulum
suspended from the bob of the other. The ``upper'' pendulum has mass $m_1$ and length $l_1$,
the ``lower'' pendulum has mass $m_2$ and length $l_2$, and  both pendulums move in the
same vertical plane. Find the Lagrangian and deduce the Lagrange's equations of motion. 
\hfill[4+4]

 \item Using Hamilton's principle show that Euler-Lagrange equation remains unchanged under the transformation 
 \[L(q, \dot{q}, t)\rightarrow L(q, \dot{q}, t) + \frac{d F(q, t)}{dt}.\]
  \hfill[2]
    
\end{enumerate}
\vspace{3.5cm}

\parskip = 10pt

\begin{center}
\large{\textbf{Indian Institute of Information Technology Allahabad}}
\end{center}
\vspace{-25pt}
\begin{center}
\large{\textbf{Quiz I}}
\end{center}
\hfill \footnotesize{Date: 19/09/2017}
\vspace{-18pt}

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

\begin{center}
\footnotesize{Paper Code: SPHY132C}
\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 Assume a frictionless, massless pulley of radius $R$ and a massless rope of length $l$. Two masses $m_1$ and $m_2$ are attached at the two ends of the rope. Let at an instant, the coordinates of the masses are $(-x_1, y_1)$ and $(x_1,y_2)$ respectively (with respect to a Cartesian coordinate system erected at the centre of the pulley). What constraint (equation) does this system of masses satisfy? What type of constraint (holonomic or non-holonomic) is this? Is the constraint sceleronomous or rheonomous? What is the number of degrees of freedom of this system? Compute the Lagrangian of the system. Hence deduce the Hamiltonian of the system. Find out the equation of motion. \hfill[1+1+1+1+3+2+1]

\item A double plane pendulum consists of two simple pendulums, with one pendulum
suspended from the bob of the other. The ``upper'' pendulum has mass $m_1$ and length $l_1$,
the ``lower'' pendulum has mass $m_2$ and length $l_2$, and  both pendulums move in the
same vertical plane. Find the Lagrangian and deduce the Lagrange's equations of motion. 
\hfill[4+4]

 \item Using Hamilton's principle show that Euler-Lagrange equation remains unchanged under the transformation 
 \[L(q, \dot{q}, t)\rightarrow L(q, \dot{q}, t) + \frac{d F(q, t)}{dt}.\]
  \hfill[2]
    
\end{enumerate}
\end{document}