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- Problem 2d The graph shows the position of a harmonic oscillator with mass m 0.700 kg on the x-axis. What is the phase constant of this oscillator? Assume the form r Acos(wt + 0 t (s) This problem does not use rounded numbers. Use the graph to derive the most accurate numbers possible. Marking will take into account the fact that you cannot read the graph with infinite precision. Show your work!dont use chat gpt!! thank youConsider 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…
- 9:50 1 ll LTE 4 A on.masteringphysics.com C ACoupled Harmonic Oscillators X2 :0 eeeee Leeee NM ) Write down the 2nd law for each of the masses. Use coordinates x, aud x, for m and M, and i,, *2 and a, a2. Hirnt: the force from spring I on monly depends on x, , but the ferce from spring 2 on m depand s on (x2-x,) two differantial 2) To simplify, let k, -k,k and m-M. Rewrite egg equatious from i) right above each other. your 3) Detine two new variables, X (Greek"chi") - x,+X2 and Ax = X,-x 2 two di ferential equations to produce Then add and subtract your very simple (harmanic) ones. 4) Write down the solutiou to bothe equations using Wask+2k2 ond evefficients A, B, C, and D two new , but w. : un 5) Now assune k-10k2 (this means that the two masses are "weakly coupled"). Also assume X,(0) = -10cm, x,(0)=0, xz (0)= o, x,(0)=0. Solve for A,B, C, D, and solve for x (t).Problems: 1. Find the Lagrange equations of motion for a simple pendulum, with length l, hung - from the ceiling of an elevator which is accelerating upwards with a.
- solved several similar problem s before, so here's the harmonic o scillator ver sion. PROBLEM 4: A SPRING AND A MASS A spring with a spring constant k is initially resting on the floor. A persn then comes over and compresses the spring by putting a mass m on the spring and pushing it a distance A. That is A is the compression dist an ce of the spring and the amount it has been displaced from equilibrium (as shown in the figure above). Now, dearly, for some (large) values of A the spring will laun ch the m ass m up as a projectile. For other (small) values of A the sy stem will oscillate. Draw a free body diagram of the system and cal cul ate A. The frequency of oscill ations of the spring, and B. For what values of A will the mass be launched upw ar d like a projectile? MunnTO Graph 3. Set the frequency to 25% as shown by the blue arrow and the amplitude to 50%, as shown by the red arrow. Then let the O Normal O Slow simulation run for some cycles. 4. Then increase the frequency through 50% and 75%. Let them run for several cycles so you can observe the patterns. Question: What do you notice happens? Please describe your observations at least two sentences.function of r. Consider a small displacement r= Ro+r' and use the binomial theorem: (1+x)" = 1+ nx + 2! п(п — 1) n(n — 1)(n — 2) з + -x' + 3! to show that the force does approximate a Hooke's law force. 71. Consider the der Waals potential van 12 U(r) = U. used model the to potential energy function of two molecules, where the minimum potential is at r = Ro. Find the force as a