Physics Cup – TalTech 2021 – Pr. 3, Hint 3.

This will be the last hint for Problem 3, so it will be fairly detailed. The final deadline for the submissions is tentatively 4th April, 13:00 GMT, with a possible extension by one more week

First, the recommended parametrization is the rotation angle of the bars, $\varphi_1,\ldots \varphi_8$ which will be our generalized coordinates. Assuming the center of mass is at the origin, these angles fully describe the state of the system, so these are valid generalized coordinates. By fixing the center of mass to the origin, we have eliminated the two translational modes where the whole system moves either in the $x$– or $y$-direction with a constant velocity. However, not all the combinations of the angles $\varphi_1,\ldots \varphi_8$ are valid as in the chain of bars, the endpoint of the last bar needs to coincide with the starting point of the first bar, so one vector constraint (or two scalar constraints) needs to be satisfied. This constraint can be written conveniently in terms of complex numbers: $\mathrm e^{\mathrm i\varphi_1} + \mathrm e^{\mathrm i(\pi/4+\varphi_2)}+\ldots + \mathrm e^{\mathrm i(7\pi/4+\varphi_8)}=0$. We consider small oscillations, so the angles $\varphi_i$ are small and we can linearize: ${\varphi_1} + e^{\mathrm i\pi/4}\varphi_2+\ldots + \mathrm e^{\mathrm i7\pi/4}\varphi_8=0$. This condition can be geometrically interpreted so that the sum of eight vectors of lengths $\varphi_i$, each next rotated with respect to the previous one by ${45}\degree$, must be zero. This is a linear vector constraint that defines a six-dimensional subspace of the eight-dimensional coordinate space $\varphi_i$.

With the coordinates $\varphi_i$, the mode shown in the leftmost figure of the second hint is described by the eigenvector $\vec \Phi_1=(1,0,-1,0,1,0,-1,0)$ – this means that the angles change in time as $\vec \Phi\equiv (\varphi_1(t), \varphi_1(t),\ldots)=(1,0,-1,0,1,0,-1,0)\varphi(t)=\varphi(t)\vec\Phi_1$. We can easily check that the constraint given above is also satisfied (although we don’t need to check it as we deduced this eigenvector from the figure of a valid oscillation mode, and therefore the constraint is satisfied by construction): in the vector diagram of the constraint, the first and the fifth vectors have the same length, but the vectors are rotated by 180 degrees, so they cancel out; the third and seventh vectors cancel out in the same way. It is easy to see that, there is another mode $\vec \Phi_2=(0,-1,0,1,0,-1,0,1)$ which is clearly linearly independent (it is even orthogonal as $\vec \Phi_2\cdot \vec \Phi_1=0$), and due to symmetry, it must have the same oscillation frequency $\omega_1$. However, the mode $\vec \Psi_1=(-1,0,1,0,-1,0,1,0)$ depends linearly on $\cdot\vec \Phi_1$ as $\cdot \Psi_1=-\Phi_1$. It is also interesting to notice that the mode $\vec \Phi_1+\Phi_2=(1,-1,-1,1,1,-1,-1,1)$) is also a natural mode of frequency $\omega_1$ as it is a superposition of two modes of the same frequency. However, it is now a mode linearly dependent on $\Phi_1$ and $\Phi_2$.

If you managed to get a proper inspiration from the rightmost figure of the Hint 2 with ring oscillations, you might have managed to get another natural mode with eigenvector $\vec \Phi_3=(1,-1,1,-1,1,-1,1,-1)$. According to general theory (eigenvectors corresponding to different eigenvalues of a symmetric matrix must be orthogonal), $\vec \Phi_3\cdot \vec \Phi_1=\vec \Phi_3\cdot \vec \Phi_2=0$. Furthermore, there is the trivial mode where the whole octagon rotates with constant speed, $\vec \Phi_0=(1,1,1,1,1,1,1,1)$ which is orthogonal to $\vec \Phi_1$, $\vec \Phi_2$, and $\vec \Phi_3$ – as it needs to be. So, we have found four eigenvectors in the six-dimensional subspace (defined by the vector sum constraint), and are missing only two eigenvectors which must be orthogonal to each of the eigenvectors found thus far. Just by trial and error, it shouldn’t be too difficult to figure out what these vectors must be.

The interesting part of Problem 3 basically ends when you have found all the eigenvectors: what remains to be done is writing down carefully the kinetic and potential energies, expressed in terms of $\dot\varphi$ and $\varphi$, for each of the modes $\vec \Phi(t)=\varphi(t)\vec \Phi_i$, and obtain the frequencies as described with the Hint 1. The expressions for potential energies will be simple, but be careful with kinetic energies: the bars can both rotate, and move translationally.

Please submit the solution of this problem via e-mail to physcs.cup@gmail.com.

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