# Question 1. (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. At what forcing frequencies does the benzene ring resonate under the photon irradience? Using various matrices, but without performing any detailed algebra or computing any inverses, find the general solution of the problem when resonance does not occur. Highlight the problem with this formal solution when the forcing frequency is resonant.

Transcribed Image Text: 1. (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. At what forcing frequencies does the benzene ring resonate under the photon irradience? Using various matrices, but without performing any detailed algebra or computing any inverses, find the general solution of the problem when resonance does not occur. Highlight the problem with this formal solution when the forcing frequency is resonant.
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Transcribed Image Text: 1. (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. At what forcing frequencies does the benzene ring resonate under the photon irradience? Using various matrices, but without performing any detailed algebra or computing any inverses, find the general solution of the problem when resonance does not occur. Highlight the problem with this formal solution when the forcing frequency is resonant.
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/1SOLUTION We are given H atoms are ignored.In infrared spectroscopy, units called wavenumbers are normally used to denote different types of light. The frequency, wavelength, and wavenumber are related to each other via the following equation:$$\mathrm{{c}={v}\lambda}$$$$\mathrm{{W}=\frac{{1}}{{\lambda}}}$$ (1)W= wave numberThese equations show that light waves may be described by their frequency, wavelength or wave number. Here, we typically refer to a light wave's frequency or wavelength.When a molecule absorbs infrared radiation, it's chemical bonds vibrate. The bonds can stretch, contract, and bend. This 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) from as follows: \( \mathrm{\fra ... See the full answer