14.E: Coupled linear oscillators (Exercises)
- Page ID
- 14261
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Two particles, each with mass \(m\), move in one dimension in a region near a local minimum of the potential energy where the potential energy is approximately given by \[U = \frac{1}{2} k (7x^2_1 + 4x^2_2 + 4x_1x_2)\nonumber\] where \(k\) is a constant.
- Determine the frequencies of oscillation.
- Determine the normal coordinates.
2. What is degeneracy? When does it arise?
3. The Lagrangian of three coupled oscillators is given by: \[\sum^3_{n=1} \left[\frac{m\dot{x}^2_n}{2} - \frac{kx^2_n}{2} \right] + k^{\prime}(x_1x_2+x_2x_3).\nonumber\] Find \(x_2(t)\) for the following initial conditions (at \(t = 0\)): \[(x_1, x_2, x_3)=(x_0, 0, 0), :: (\dot{x}_1, \dot{x}_2, \dot{x}_3) = (0, 0, v_0). \nonumber\]
4. A mechanical analog of the benzene molecule comprises a discrete lattice chain of 6 point masses \(M\) connected in a plane hexagonal ring by 6 identical springs each with spring constant \(\kappa\) and length \(d\).
- List the wave numbers of the allowed undamped longitudinal standing waves.
- Calculate the phase velocity and group velocity for longitudinal travelling waves on the ring.
- Determine the time dependence of a longitudinal standing wave for a angular frequency \(\omega = 2\omega_{cutoff}\), that is, twice the cut-off frequency.
5. Consider a one dimensional, two-mass, three-spring system governed by the matrix \(A\), \[A = \begin{pmatrix} 4 & -2 \\ -2 & 7 \end{pmatrix}\nonumber\] such that \(Ax = \omega^2x\),
- Determine the eigenfrequencies and normal coordinates.
- Choose a set of initial conditions such that the system oscillates at its highest eigenfrequency.
- Determine the solutions \(x_1(t)\) and \(x_2(t)\).
6. Four identical masses \(m\) are connected by four identical springs, spring constant \(\kappa\), and constrained to move on a frictionless circle of radius \(b\) as shown on the left in the figure.
- How many normal modes of small oscillation are there?
- What are the eigenfrequencies of the small oscillations?
- Describe the motion of the four masses for each eigenfrequency.
7. Consider the two identical coupled oscillators given on the right in the figure assuming \(\kappa_1 = \kappa_2 = \kappa\). Let both oscillators be linearly damped with a damping constant \(\beta\). A force \(F = F_0 \cos(\omega t)\) is applied to mass \(m_1\). Write down the pair of coupled differential equations that describe the motion. Obtain a solution by expressing the differential equations in terms of the normal coordinates. Show that the normal coordinates \(\eta_1\) and \(\eta_2\) exhibit resonance peaks at the characteristic frequencies \(\omega_1\) and \(\omega_2\) respectively.
8. As shown on the left below the mass \(M\) moves horizontally along a frictionless rail. A pendulum is hung from \(M\) with a weightless rod of length \(b\) with a mass \(m\) at its end.
- Prove that the eigenfrequencies are \[\omega_1 = 0 \quad \omega_2 = \sqrt{\frac{g}{Mb} (M + m)} \nonumber\]
- Describe the normal modes.
