Consider the dispersion relation of a linear spiral density wave perturbation (equation 4.45 in Chapter 3 of the lecture notes). The Toomre's parameter is(V2)²Kdefined as Q = =1/2лGσ。,where (V2) 1/2 is the root-mean-square turbulent plus thermal velocity, the epicyclic frequency, σ the unperturbed surfacedensity of the disc and G the gravitational constant. If Q = 0.7, for each of the the following values of the radial wavenumber k choose whether theperturbation makes the spiral pattern stable or unstable.kk=II=24KK1/2 stable(1/2 stable(V2)1/2Κk = (V unstable2 1/2÷→k = 2.K(V2)1/2unstable →k =K(V2)1/2unstable →

Step 1: Understanding the Toomre Q ParameterThe Toomre Q parameter is a unitless value that…

Answered: Consider the dispersion relation of a…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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= (V2)1/2 K πGOO ' Consider the dispersion relation of a linear spiral density wave perturbation (equation 4.45 in Chapter 3 of the lecture notes). The Toomre's parameter is defined as Q where (V2) 1/2 is the root-mean-square turbulent plus thermal velocity, к the epicyclic frequency, σ the unperturbed surface density of the disc and G the gravitational constant. For Q = 1/√2, show that the spiral pattern is unstable if К (√2-1) (V²² ) 1/2 < |k| < (√√2+1) where k is the wavenumber of the perturbation. К (V+2)1/2
ii. The velocity potential for a two dimensional flow is O = x(2y - 1). Determine the velocity at the point P(4, 5). Also obtain the value of stream function at P.
write the solution step by step and clearly.
Consider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on. A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators? B) Obtain the average energy of N oscillators at temperature T from the partition function. C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this? D) Find the Helmholtz free energy from this system. E) Derive the expression that gives the entropy of this system for the temperature.
Problem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?
Consider an ideal gas containing N atoms in a container of volume Pressure P, and absolute temperature T1 (not to be confused with K. E. T). Use the virtual theorem to derive the equation of state for a perfect gas.
The Newton–Raphson method for finding the stationary point(s) Rsp of a potential energy surface V is based on a Taylor-series expansion around a guess, R0. Similarly, the velocity Verlet algorithm for integrating molecular dynamics trajectories [xt, vt] is based on a Taylor-series expansion around the initial conditions, [x0, v0]. Both of these methods rely strictly on local information about the system. How many derivatives do we need to compute in order to apply them?
Find the number density N/V for Bose-Einstein condensation to occur in helium at room temperature (293 K). Compare your answer with the number density for an ideal gas at room temperature at 1 atmosphere pressure.
Please help me in a step by step solution for problem #10?
When a river flows at a velocity V past a circular pylon of diameter D, vortices are shed at a frequency f. It is known that f is also a function of the water density ρ and viscosity µ, and the acceleration due to gravity, g. (a) Use dimensional analysis to express this information in terms of a functional dependence on nondimensional groups. (b) A test is to be performed on a 1/4th scale model. If previous tests had shown that viscosity is not important, what velocity must be used to obtain dynamic similarity, and what shedding frequency would you expect to see?
2b. Consider that the input to the tank-pipe liquid system is formed of a source pressure pi, the atmospheric pressure pa, and the input flow rate q., whereas the output is the output flow rate q. Derive the corresponding transfer function matrix by means of complex impedances considering that known are the element properties Ci and R. Utilize the transfer function approach using a) matlab and b)Simulink to plot the output flow rate of the liquid system shown. Known are pa = 10 N/m, pi =2.2x10 N/m2, q=0.03 m3/s. d=1.5 m (diameter of the tank). p= 1000 kg/m', d, = 0.020 m (diameter of the pipe). / = 18 m (length of pipe), u=0.001N s/m?. P. Tank Pipe R. Pressure source
Consider a non-rotating circular thin disc of gas of radius R. The only forces present in the system are pressure forces within the disc and its self-gravity. The disc is surrounded by empty space. In the disc is present a surface density perturbation of the type 01 = 010e (wt-kr) where σ10 is the amplitude of the perturbation, t represents time, r the radial coordi- nate from the centre of the disc, w is the angular frequency of the perturbation and k its wavenumber. Under the influence of the above perturbation, the linear stability of the disc is determined by the following dispersion relation w² = u²k² - 2πGook, where u is the sound speed in the disc, σ the surface density of the disc, and G is the gravitational constant. 1. Using the dispersion relation and appropriate definitions derive an expression of the group velocity of the small perturbations as a function of u, σo and their wavelength. 2. State the criterion for the disc to be stable and then show that the disc is stable…

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