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\documentclass[11pt]{beamer}
\usetheme{Ilmenau}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usetheme{default}
\usepackage{float}
\begin{document}
    \author{Anshul Sharma}
    \title{Forced Oscillator}
    \institute {CENTRAL UNIVERSITY OF HIMACHAL PRADESH}

    \begin{frame}[plain]
    \maketitle
\end{frame}

\begin{frame}[t]
\frametitle{Contents}
\begin{enumerate}\Huge
    \item  Aim
    \item Objectives
    \item Apparatus
    \item Theory
    \item Experimental Setup
    \item Observations
    \item Results

\end{enumerate}
\end{frame}
\begin{frame}
\frametitle{Aim}\Large
To understand the motion of the system when being driven by an external force sinusoidally.
\end{frame}
\begin{frame}[t]\vspace{10pt}
\frametitle{Objectives}
\begin{enumerate}\huge
    \item To study the Amplitude Resonance curve at variour frequencies.
    \item To determine the quality factor of our given system (i.e.  Spoke).
\end{enumerate}
\end{frame}
\begin{frame}[t]\vspace{10pt}
\frametitle{Apparatus}\huge
\begin{enumerate}
    \item Exp-Eyes
    \item DC motor
    \item Spoke
    \item Meter Scale
\end{enumerate}
\end{frame}
\begin{frame}
\frametitle{Theory}\Large
\textbf{Forced Oscillator}: When a system is subjected to an external force, (say periodic) the oscillator will exhibit Forecd oscillations and is called Forced Oscillator.
\end{frame}
\begin{frame}[t]
\frametitle{Equation of motion for Forced Oscillator}
If a system is given an force to oscillate as long as the driven frequency $\omega$ is given to it, then by Newton 2nd law\\
\begin{equation*}
m\ddot{x}+2m\mu\dot{x}+\k2x=f_0(t)
\end{equation*}
or
\begin{equation*}\ddot{x}+2\mu\dot{x}+\omega_0^2x=\frac{f_0(t)}{m} \end{equation*}
where \textbf{m} is the mass of the system being oscillated.\\
\textbf{$\mu$} is the damping coefficent.\\
\textbf{$\omega_0^2$} is the natural frequency of the system.\\
\textbf{$f_0$} is the external force being applied to the system.
\end{frame}
\begin{frame}[t]
If the external force being applied is sinusoidal we have,\\
\begin{center}
    f=$f_0\sin\omega t$ \\
\end{center}
     Then our equation of motion becomes,\\
\begin{equation*}\ddot{x}+2\mu\dot{x}+\omega_0^2x=\frac{f_0\sin\omega t}{m} \end{equation*}
\textbf{$\mu$} is the damping coefficent.\\
\textbf{$\omega_0^2$} is the natural frequency of the system.\\
\textbf{$\omega^2$} is the driven frequency supplied to the system.\\
\textbf{$f_0$} is the external force being applied to the system.\\
This is a linear nonhomogenous differential equation.
\end{frame}
\begin{frame}
The solution of the equation is assumed as,\\
\begin{equation*} x=A\sin(\omega t-\theta). \end{equation*}
where \textbf{A} is the amplitude and \textbf{$\theta$} is the phase angle.\\
On solving the differntial equation we get,\\
\begin{equation*}  A = 
\frac{f_0/m}{\sqrt{(\omega^{2}-\omega_{0}^2)^2+4\mu^{2}\omega^{2}}} \end{equation*} and \\
\begin{equation} \tan{\theta} = 
\frac{2\mu\omega}{(\omega^{2}-\omega_{0}^2)} \end{equation}.
\end{frame}
\begin{frame}[t]
As resonance occurs at,\\
\begin{center}
$\omega_{r}$ $\sim$ $\omega_{0}$
\end{center}
Therefore resonant frequency become,
\begin{center}
    \begin{equation*} \omega_r = \sqrt{\omega_{0}^2-2\mu^2} \end{equation*}
    \end{center} 
And the amplitude becomes,\\
\begin{equation} A_{max} = \frac{f_0/m}{2\mu\sqrt{\omega_0^2-\mu^2}} 
\end{equation}
\end{frame}
\begin{frame}[t]
\frametitle{Quality Factor}
\textbf{Quality Factor}: It is defined as 2$\pi$ times the ratio of the average energy stored in the system to the energy dissipated per cycle by the applied force.
Where for forced oscillator the quality factor is,\\
\begin{equation} Q=\frac{\omega_{0}}{2\pi} \end{equation}.
\end{frame}
\begin{frame}[t]
\frametitle{Quality Factor}
It can also be calculated by the amplitude resonance curve as,
\begin{equation} |\omega_{h}-\omega_r|=\Delta=\mu\sqrt{3} 
\end{equation}.
This shows that for small $\mu$, $\omega_{h}$ is close to $\omega_r$, 
$\Delta$ is small and the resonance is sharp.\\
\begin{figure}
    \centering
    \includegraphics[width=0.5\linewidth]{"C:/Users/vinay kumar/Desktop/latex work/IMG_20180518_111515"}
    \caption{}
    \label{fig:img20180518111515}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Experiment Setup}
Experiment Setup Screenshot
\begin{figure}
    \centering
    \includegraphics[width=0.7\linewidth]{../screenshot001}
    \caption{}
    \label{fig:screenshot001}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Calculating Natural Freqeuncy and damping factor}
Calculation of natural frequency and damping factor by Exp-Eyes Screenshot.\\
1) \begin{figure}
    \centering
    \includegraphics[width=0.9\linewidth]{../screenshot002}
    \caption{}
    \label{fig:screenshot002}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Calculating Natural Freqeuncy and damping factor}
2) \begin{figure}
    \centering
    \includegraphics[width=0.7\linewidth]{../screenshot003}
    \caption{}
    \label{fig:screenshot003}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Applying differnet frequency from Exp-Eyes}
Screenshot\\
1) \begin{figure}
    \centering
    \includegraphics[width=0.7\linewidth]{../screenshot004}
    \caption{}
    \label{fig:screenshot004}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Applying differnet frequency from Exp-Eyes}
Screenshot\\
2) \begin{figure}
    \centering
    \includegraphics[width=0.5\linewidth]{../screenshot005}
    \caption{}
    \label{fig:screenshot005}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Observations}
 \textbf{1)Frequency and damping coefficent of oscillator from expeyes} \\ \\
\begin{center}
\begin{tabular}{|c|c|}
    \hline
    \textbf{Frequency} &\textbf{Damping Factor} \\
    \hline
    1.56&0.536 \\
    \hline
    1.56&0.476 \\
    \hline
    1.56&0.402 \\
    \hline 
    1.56&0.476 \\
    \hline
    1.56&0.387 \\
    \hline
    1.57&0.410 \\
    \hline
    1.56&0.407 \\
    \hline
    1.56&0.345 \\
    \hline
    1.56&0.391 \\
    \hline
    1.55&0.412 \\
    \hline  
    Mean=1.56$\pm$0.01&0.424$\pm$0.001 \\
    \hline
\end{tabular}
\end{center}
\end{frame}
\begin{frame}
\textbf{2) Variation in frequency from tracker and expeyes}\vspace{8pt}
\begin{tabular}{|c|c|}
    \hline
    \textbf{Frequency from expeyes}&\textbf{Frequency from Tracker} \\
    \hline
    1.20&1.22 \\
    \hline
    1.22&1.24 \\
    \hline
    1.28&1.24 \\
    \hline
    1.32&1.32 \\
    \hline
    1.36&1.36\\
    \hline
    1.44&1.46 \\
    \hline
    1.50&1.49 \\
    \hline
    1.51&1.50 \\
    \hline
    1.52&1.56 \\
    \hline
    1.57&1.59\\
    \hline
    1.64&1.62 \\
    \hline
    1.74&1.74\\
    \hline
\end{tabular}
\end{frame}
\begin{frame}
\textbf{3)Amplitude(calculated from tracker) for various frequencies from waveform generator}\vspace{8pt}
\begin{tabular}{|c|c|}
    \hline
    \textbf{Frequency ($\nu$) In Hz}&\textbf{Amplitude (A) in cm} \\
    \hline
    1.22&7.38 \\
    \hline
    1.24&7.61 \\
    \hline
    1.29&9.16 \\
    \hline
    1.32&10.53 \\
    \hline
    1.36&12.00 \\
    \hline
    1.46&15.75 \\
    \hline
    1.49&24.82 \\
    \hline
    1.50&25.89 \\
    \hline
    1.56&23.52 \\
    \hline
    1.59&6.39 \\
    \hline
    1.62&5.53 \\
    \hline
    1.74&4.12 \\
    \hline
\end{tabular}
\end{frame}
\begin{frame}[t]
\frametitle{Results}\huge
\begin{enumerate}
    \item Theoretical natural Frequency of the Spoke=(1.570$\pm$0.004)Hz.
    \item Experimental natural Frequency of the spoke=(1.50$\pm$0.05)Hz.
    \item Theoretical quality factor of the spoke=
    \item Experimental quality factor of spoke=
\end{enumerate}
\end{frame}
\end{document}
  • Welcome to TeX.SE! Note that in oder to mark the code as code, just take it into the cursor and press the {} above the text field when you post or edit. I believe the issue is the double \\\ \\` after \textbf{1)Frequency and damping coefficent of oscillator from expeyes}`. There are much more elegant ways to achieve vertical spacing, if that's the reason why you had it. – user121799 May 25 '18 at 2:51
  • You have 21 frames here. Could you give us just the frame that causes the error? – Teepeemm May 25 '18 at 2:54
  • Welcome to TeX.SE! Please inform yourself about MWEs :D – thymaro May 25 '18 at 5:25
  • 1
    Since you have some responses below that seem to answer your question, please consider marking one of them as ‘Accepted’ by clicking on the tickmark below their vote count (see How do you accept an answer?). This shows which answer helped you most, and it assigns reputation points to the author of the answer (and to you!). It's part of this site's idea to identify good questions and answers through upvotes and acceptance of answers. – samcarter_is_at_topanswers.xyz May 25 '18 at 15:48
3

The problem also occurs for this drastically cut down document:

\documentclass{beamer}

\begin{document}

\begin{frame}
  \frametitle{Observations}
  \textbf{1)Frequency and damping coefficent of oscillator from
  expeyes} \\ \\
\end{frame}

\end{document}

It is caused by \\ \\. The beamer package redefines \\ so that it can not be used on a blank line; the first \\ creates a newline, the next is then on that blank line.

If you need larger space here, either use something like

\\[2ex]

where the 2ex gives 2ex more vertical space, or better

\medskip

(or \bigskip) after a blank line. Cf. What does double backslash in LaTeX mean?

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