Physics Of Oscillations In Physics

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This report investigates resonance phenomena of driven damped simple harmonic oscillations by measuring accelerations and frequencies of an oscillator. Since it turned out that oscillations with low damping have smaller errors than those with high damping, small damping cases are employed to examine the resonance phenomena. Based on the Newton's 2nd Law \cite{book}, the motion of oscillations can be modeled by the following differential equation and its solution.
\begin{equation} \label{eq1}
\ddot x+2\gamma \dot x+\omega_{0}^{2} x=\left(\frac{F_{0}}{m}\right)e^{i\omega t}
\end{equation}
\begin{equation} \label{eq2} x(\omega)=\frac{F_{0}/m}{\omega_{0}^{2}-\omega^{2}+2i\gamma\omega}=A(\omega)cos(\omega t-\phi)
\end{equation}
Acceleration
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In total, 16 experimental data points have been taken after settling to each steady-state for various driven frequencies. The voltage data obtained in Lab View is then converted to the acceleration data by multiplying its amplitudes by the slope of the two calibration points. Here, the voltage data is calibrated with gravity only and with no acceleration. The slope is called calibration factor. Errors for accelerations are the calibration factor times the error of the cursor measurements of Lab View, and that is the minimum difference Lab View can measure. Errors for the phase shifts are obtained by equation \ref{eq5} where \(\Delta f¥\) is an error for the driven frequency derived from the variation in frequency displayed on the oscilloscope \cite{oscilloscope}, \(t\) is a time shift between the peak of the acceleration of the oscillator and the driven force, \(\Delta t\) is an error for the time derived from the grid square size in the Lab View waveform graph.
\begin{equation} \label{eq5}
\Delta \phi = \phi \sqrt{(\frac{\Delta f}{f})^{2}+(\frac{\Delta t}{t})^{2}}
\end{equation}
At the beginning of the experiment, the natural frequency of the oscillator with neither the eddy currents nor driving force is observed as \(\omega_{0} = 75.7 \pm 0.1 \) rad/s or equivalently \(f_{0} = 12.05 \pm 0.01 \) Hz.

The result of the experiment is summarized in Figure ~\ref{fg:A} and ~\ref{fg:AandPhi}. The overall

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