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Tutorial 6 PHY115 PHY115 OSCILLATIONS AND WAVES Tutorial 6 (10/09/2024) Instructor: Sivasurender Chandran 1. In the class, we have seen the synchronization of metronomes when...

Tutorial 6 PHY115 PHY115 OSCILLATIONS AND WAVES Tutorial 6 (10/09/2024) Instructor: Sivasurender Chandran 1. In the class, we have seen the synchronization of metronomes when kept on a platform standing on the two water bottles (see this video, especially from 11:40 to 14:11, to remind yourself). To model this system, we will set up a coupled oscillator of the form shown below, where the dashpot indicates damping force. Using this, we will show why only symmetric motion survives after long enough time. 2. A coupled pendulum is comprised of three masses and two springs as shown in the figure here. Considering the rest length of the springs to be same, deduce the resonant frequencies of all the normal modes. 3. The figure below represents a triatomic molecule with a heavy atom (with mass M) bound to two identical atoms of smaller mass m on either side. The binding is represented by springs of stiffness k and in equilibrium the atom centres are equally spaced along a straight line. Simple harmonic oscillations are considered only along this linear axis. Set up the equation of motion for each atom and deduce the frequencies of normal modes. Practice problems (will not be discussed in the tutorial): 4. In the coupled pendulum shown here, the mass “a” is forced horizontally with a force 𝐹0 cos 𝜔𝑡. Assuming no damping, deduce the amplitudes of the normal modes and compare them with that of the individual displacements 𝑥𝑎 and 𝑥𝑏. 5. Two masses m1 and m2 are coupled via a spring with stiffness k. Deduce the expression for the resonant frequency of the system? Tutorial 6 PHY115 6. The equal masses shown in the pendulum oscillate vertically. Deduce the resonant frequency of the system and the amplitude ratios of the two masses in both the resonant modes. 7. Consider the mass spring system shown here. Deduce the normal frequencies, and the corresponding normal modes of the system. 8. The figure below shows two identical LC oscillators coupled by a common capacitance C with the direction of current flows indicated by arrows. Identify the normal coordinates and normal frequencies of the system.