Find the natural frequencies and mode shapes (mode 1 and mode 2) of the system shown in Figure 1 for m₁ = m₂ = 2m and k₁ = k₂ = 2k. Base x₁(1) x₂(1) k₁ m₁ m₂ Figure 1
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- Consider a transverse periodic (sinusoidal) wave passing through a very long string of mass density 0.250 kg/m. The wave function for this wave is found to be: y (x,t) = (0.125 m) cos [(1.10 rad/m) x - (15.0 rad/s) t] From the equation find the following quantites; 1. Wave amplitude 2. Wave number 3. Angular frequency 4. Wavelength 5. Period 6. Frequency3. A system of N harmonic oscillators of frequency w are prepared in identical initial states of wavefunction Þ(x,0). It is found that the measurement of the energy of the system at t = 0 gives 0.5hw with probability 0.25, 1.5hw with probability 0.5 and 2.5hw with probability 0.25. a. Write a possible function p(x, 0). b. Write the corresponding (x, t). c. What is the expectation value of the Hamiltonian in the state p(x,t) ? d. Calculate the expectation value of position at time tThe ability to hear a "pin drop" is the sign of sensitive hearing. Suppose a 0.53 g pin is dropped from a height of 31 cm, and that the pin emits sound for 1.4 s when it lands. Part A Assuming all of the mechanical energy of the pin is converted to sound energy, and that the sound radiates uniformly in all directions, find the maximum distance from which a person can hear the pin drop. (This is the ideal maximum distance, but atmospheric absorption and other factors will make the actual maximum distance considerably smaller.) Express your answer using two significant figures. Hνα ΑΣφ ? r = km