Question This is a differential equations problem. Please do the two substitution verifications the question asks for  (9 pts) In a model of the benzene ring, the hydrogen atoms are ignored and the six carbon atoms are treated as individual particles with angular positions, \( \theta_{j}(t), j=1,2, \ldots, 6 \) and mass \( m \). Each atom is connected to its neighbour by a bond, modelled as a spring with constant \( k \), and the absorption of incoming photons applies a sinusoidal external force with frequency \( \Omega>0 \). Newton's law for the atoms states that \[ \begin{array}{l} m \ddot{\theta}_{1}=k\left(\theta_{2}-\theta_{1}\right)-k\left(\theta_{1}-\theta_{6}-2 \pi\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{2}=k\left(\theta_{3}-\theta_{2}\right)-k\left(\theta_{2}-\theta_{1}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{3}=k\left(\theta_{4}-\theta_{3}\right)-k\left(\theta_{3}-\theta_{2}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{4}=k\left(\theta_{5}-\theta_{4}\right)-k\left(\theta_{4}-\theta_{3}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{5}=k\left(\theta_{6}-\theta_{5}\right)-k\left(\theta_{5}-\theta_{4}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{6}=k\left(2 \pi+\theta_{1}-\theta_{6}\right)-k\left(\theta_{6}-\theta_{5}\right)+\alpha \sin \Omega t \end{array} \] where \( \alpha \) is the amplitude of the photon forcing. Write these ODEs as the system, \[ \ddot{\mathbf{x}}=A \mathbf{x}+\mathbf{c}+\mathbf{a} \sin \Omega t \] identifying the coefficient matrix \( A \) and the two constant vectors \( \mathbf{c} \) and \( \mathbf{a} \). Verify (by substitution and performing a matrix multiplication) that there is a time-independent particular solution of \( \theta_{j}=\frac{1}{3}(j-1) \pi \) for \( \alpha=0 \) (corresponding to the atoms sitting evenly spaced at equilibrium). Confirm (again by substitution) that \( A \) has the eigenvectors, \[ \mathbf{e}_{1}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}\right), \mathbf{e}_{2}=\left(\begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{3}=\left(\begin{array}{c} 1 \\ -1 \\ 0 \\ 1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{4}=\left(\begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{5}=\left(\begin{array}{c} 1 \\ 1 \\ 0 \\ -1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{6}=\left(\begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \\ -1 \\ -1 \end{array}\right) \] and find the corresponding eigenvalues, \( \lambda \), in terms of \( \omega=\sqrt{k / m} \). Briefly interpret to what motions of the atoms these eigensolutions correspond.

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This is a differential equations problem. Please do the two substitution verifications the question asks for 

Transcribed Image Text: (9 pts) In a model of the benzene ring, the hydrogen atoms are ignored and the six carbon atoms are treated as individual particles with angular positions, \( \theta_{j}(t), j=1,2, \ldots, 6 \) and mass \( m \). Each atom is connected to its neighbour by a bond, modelled as a spring with constant \( k \), and the absorption of incoming photons applies a sinusoidal external force with frequency \( \Omega>0 \). Newton's law for the atoms states that \[ \begin{array}{l} m \ddot{\theta}_{1}=k\left(\theta_{2}-\theta_{1}\right)-k\left(\theta_{1}-\theta_{6}-2 \pi\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{2}=k\left(\theta_{3}-\theta_{2}\right)-k\left(\theta_{2}-\theta_{1}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{3}=k\left(\theta_{4}-\theta_{3}\right)-k\left(\theta_{3}-\theta_{2}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{4}=k\left(\theta_{5}-\theta_{4}\right)-k\left(\theta_{4}-\theta_{3}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{5}=k\left(\theta_{6}-\theta_{5}\right)-k\left(\theta_{5}-\theta_{4}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{6}=k\left(2 \pi+\theta_{1}-\theta_{6}\right)-k\left(\theta_{6}-\theta_{5}\right)+\alpha \sin \Omega t \end{array} \] where \( \alpha \) is the amplitude of the photon forcing. Write these ODEs as the system, \[ \ddot{\mathbf{x}}=A \mathbf{x}+\mathbf{c}+\mathbf{a} \sin \Omega t \] identifying the coefficient matrix \( A \) and the two constant vectors \( \mathbf{c} \) and \( \mathbf{a} \). Verify (by substitution and performing a matrix multiplication) that there is a time-independent particular solution of \( \theta_{j}=\frac{1}{3}(j-1) \pi \) for \( \alpha=0 \) (corresponding to the atoms sitting evenly spaced at equilibrium). Confirm (again by substitution) that \( A \) has the eigenvectors, \[ \mathbf{e}_{1}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}\right), \mathbf{e}_{2}=\left(\begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{3}=\left(\begin{array}{c} 1 \\ -1 \\ 0 \\ 1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{4}=\left(\begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{5}=\left(\begin{array}{c} 1 \\ 1 \\ 0 \\ -1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{6}=\left(\begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \\ -1 \\ -1 \end{array}\right) \] and find the corresponding eigenvalues, \( \lambda \), in terms of \( \omega=\sqrt{k / m} \). Briefly interpret to what motions of the atoms these eigensolutions correspond.
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Transcribed Image Text: (9 pts) In a model of the benzene ring, the hydrogen atoms are ignored and the six carbon atoms are treated as individual particles with angular positions, \( \theta_{j}(t), j=1,2, \ldots, 6 \) and mass \( m \). Each atom is connected to its neighbour by a bond, modelled as a spring with constant \( k \), and the absorption of incoming photons applies a sinusoidal external force with frequency \( \Omega>0 \). Newton's law for the atoms states that \[ \begin{array}{l} m \ddot{\theta}_{1}=k\left(\theta_{2}-\theta_{1}\right)-k\left(\theta_{1}-\theta_{6}-2 \pi\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{2}=k\left(\theta_{3}-\theta_{2}\right)-k\left(\theta_{2}-\theta_{1}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{3}=k\left(\theta_{4}-\theta_{3}\right)-k\left(\theta_{3}-\theta_{2}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{4}=k\left(\theta_{5}-\theta_{4}\right)-k\left(\theta_{4}-\theta_{3}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{5}=k\left(\theta_{6}-\theta_{5}\right)-k\left(\theta_{5}-\theta_{4}\right)+\alpha \sin \Omega t \\ m \ddot{\theta}_{6}=k\left(2 \pi+\theta_{1}-\theta_{6}\right)-k\left(\theta_{6}-\theta_{5}\right)+\alpha \sin \Omega t \end{array} \] where \( \alpha \) is the amplitude of the photon forcing. Write these ODEs as the system, \[ \ddot{\mathbf{x}}=A \mathbf{x}+\mathbf{c}+\mathbf{a} \sin \Omega t \] identifying the coefficient matrix \( A \) and the two constant vectors \( \mathbf{c} \) and \( \mathbf{a} \). Verify (by substitution and performing a matrix multiplication) that there is a time-independent particular solution of \( \theta_{j}=\frac{1}{3}(j-1) \pi \) for \( \alpha=0 \) (corresponding to the atoms sitting evenly spaced at equilibrium). Confirm (again by substitution) that \( A \) has the eigenvectors, \[ \mathbf{e}_{1}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}\right), \mathbf{e}_{2}=\left(\begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{3}=\left(\begin{array}{c} 1 \\ -1 \\ 0 \\ 1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{4}=\left(\begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{5}=\left(\begin{array}{c} 1 \\ 1 \\ 0 \\ -1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{6}=\left(\begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \\ -1 \\ -1 \end{array}\right) \] and find the corresponding eigenvalues, \( \lambda \), in terms of \( \omega=\sqrt{k / m} \). Briefly interpret to what motions of the atoms these eigensolutions correspond.
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【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/3Please My Answer PleaseWe are given that H atoms are ignored .In infrared spectroscopy, units called wave numbers are normally used to denote different types of light. The frequency, wavelength, and wave number are related to each other via the following equation: \( \mathrm{{c}=\nu\lambda} \) \( \mathrm{{W}=\frac{{1}}{\lambda}\ldots\ldots{\left({1}\right)}} \)W= wave numberThese equations show that light waves may be described by their frequency, wavelength or wave number. Here, we typically refer to light waves by their wave number, however it will be more convenient to refer to a light wave's frequency or wavelength.  When a molecule absorbs infrared radiation, its chemical bonds vibrate. The bonds can stretch, contract, and bend. ExplanationPlease Refer Next Step Explanation:Please refer to solution in this step.Step2/3Please My Answer PleaseThis is why infrared spectroscopy is a type of vibrational spectroscopy. The complex vibrational motion of a molecule can be broken down into a number of constituent vibrations called normal modes. The first necessary condition for a molecule to absorb infrared light is that the molecule must have a vibration during which the change in dipole moment with respect to distance is non-zero. This condition can be summarized in equation(2) form as follows: \ ... See the full answer