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\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}
答案1
这个被大幅删减的文档也出现了同样的问题：
\documentclass{beamer}
\begin{document}
\begin{frame}
\frametitle{Observations}
\textbf{1)Frequency and damping coefficent of oscillator from
expeyes} \\ \\
\end{frame}
\end{document}
这是由 引起的\\ \\。beamer包重新定义\\，因此它不能在空白行上使用；第一个\\创建一个换行符，下一个则在该空白行上。
如果你需要更大的空间，可以使用类似
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提供2ex更多2ex垂直空间，或更好
\medskip
（或\bigskip）在空白行后。参见LaTeX 中的双反斜杠是什么意思？