14.4: Center of Mass Oscillations
- Page ID
- 9638
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Transforming the coordinates into the center of mass of the two oscillating masses elucidates an interesting feature of the normal modes for the two-coupled linear oscillator. As illustrated in Figure \((14.2.1)\), the center-of-mass coordinate for the two mass system is
\[\begin{align*} 2R_{cm} &= l + x_1 + l + l^{\prime} + x_2 \\[4pt] &= 2l + l^{\prime} + \eta_2 \end{align*}\]
while the relative separation distance is
\[r = (l + l^{\prime} + x_2) − (l + x_1) = l^{\prime} − \eta_1\notag\]
That is, the two normal modes are
\[\begin{align} \eta_1 = l^{\prime} − r \\ \eta_2 = 2 R_{cm} − 2l − l^{\prime} \notag\end{align}\]
The \(\eta_1\) mode, which has angular frequency \(\omega_1 = \sqrt{\frac{\kappa +2\kappa^{\prime}}{M}}\) corresponds to an oscillations of the relative separation \(r\), while the center-of-mass location \(R_{cm}\) is stationary. By contrast, the \(\eta_2\) mode, with angular frequency \(\omega_2 = \sqrt{\frac{\kappa}{M}}\) corresponds to an oscillation of the center of mass \(R_{cm}\) with the relative separation \(r\) being a constant.
Figure \(\PageIndex{1}\) illustrates the decoupled center-of-mass \(R_{cm}\), and relative motions \(r\) for both normal modes of the coupled double-oscillator system. The difference in angular frequencies and amplitudes is readily apparent. It is of interest to consider the special case where the spring constant \(\kappa = 0\) for the two outside springs. Then the angular frequencies are \(\omega_1 = \sqrt{\frac{2\kappa^{\prime}}{M}}\) and \(\omega_2 = 0\) for the two normal modes. When \(\kappa = 0\) the \(\eta_2\) mode is a spurious center-of-mass mode since it corresponds to an oscillation with \(\omega_2 = 0\) in spite of the fact that there are no forces acting on the center of mass. That is, the center-of-mass momentum must be a constant of motion. This spurious center-of-mass oscillation is a consequence of measuring the displacements \((x_1, x_2)\) with respect to an arbitrary external reference that is not related to the center of mass of the coupled system. Spurious center-of-mass modes are encountered frequently in many-body coupled oscillator systems such as molecules and nuclei. In such cases it is necessary to project out the center-of-mass motion to eliminate such spurious solutions as will be discussed later.