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Strongly nonlinear oscillators 130 marks] Most of the nonlinear oscillators YOu studlied in lectures are weakly nonlinear. That is the restoring force is linear to...

Question

Strongly nonlinear oscillators 130 marks] Most of the nonlinear oscillators YOu studlied in lectures are weakly nonlinear. That is the restoring force is linear to leading order plus small nonlinear correction. (The relaxation oscillator, which Can serve a5 model of the human heart , is an exception:) In this question; we consider strongly nonlinear oscillators where the restoring force is something other than linear to leading order, i.e. the nonlinearity is Iot just sInall correction: Conside

Strongly nonlinear oscillators 130 marks] Most of the nonlinear oscillators YOu studlied in lectures are weakly nonlinear. That is the restoring force is linear to leading order plus small nonlinear correction. (The relaxation oscillator, which Can serve a5 model of the human heart , is an exception:) In this question; we consider strongly nonlinear oscillators where the restoring force is something other than linear to leading order, i.e. the nonlinearity is Iot just sInall correction: Consider one-dimensional oscillator of mass m described by generalised coordinate q(t) (units: m) ald generalised momentum p(t) (units: kg In 1). Suppose that the restoring force has magnitude klqla - where 0 is not necessarily an integer What are the SI units of k? Starting from the Lagrangian, show that thc oscillator's Hamiltonian takes thc form H(g,p) 2mn Suppose that the oscillator is isolated from its envirOlment, so that it vibrates with constant mnechanical energy; E Sketch its phase-space trajectory on the q P plane Label the intercepts_ Explain wby the phase space trajectory is closed for all positive values of You may find it useful to illustrate yOur answer with diagram canonical transformation cxists which converts q(t) and p(t) to action and angle variables J(t) and 0(t) respectively. Show that_ when E is constant, one obtains 1/ a J = (8rmnE)1/2 38+} where I(z) symbolises the Euler gamma functiol; with the properties I(1) = ad C(I+l) = IC(c): You may find the integrals at the end of the question helpful: Express the transformed Hamiltonian K(J,0) H[q( J,0) , p( J,8)] purely in terms of J, ie inde- pendent of 0. Write down Hamnilton $ equations for J(t) and 0(t) . It can be shown that 0(t) increases by unity during one period, of duration T_ i6 0(t+T)- 0(t) = 1 for all t. By solving Hamilton $ equations_ show that the period is given by (a +2)J 2aE with J given by (2)- By the way; You may be tempted to think that 0(t) should increase by 2v during one period; befits any decent angular coordinate! In fact; though the increment is unity. We do not prove this result here. If you are curious and would like to try proving it "for fun" (highly optional; zero marksl ) . OIlC good approach is to relate 0(+T) _ 0() = } da &q4 to the generating function for the canonical transformation that yields the action-angle variables. Now let' $ calculate the period in more traditional way. Write down Hamnilton $ equations for the original Hamniltoniall, H(g,p).



Answers

Two small speakers, 0.690 m apart, are facing in the same direction. They are driven by one 685 Hz oscillator and therefore emit identical sound waves in phase with one another at the respective points of origin. (The speed of sound waves in air is 343 m/s.)
a) A sound engineer wishes to stand in front of one of the speakers, at the closest point (i.e., smallest x-value) where intensity is at a relative maximum. At what distance x from the nearest speaker should he position himself? Note that the distance from the other speaker is greater than x. (Enter your answer in m.)

(b)The sound engineer now wishes to stand at the closest point along that line where intensity is at a relative minimum. At what distance x should he position himself now? Note that the distance from the other speaker is greater than x. (Enter your answer in m.)

Yeah, Okay, thank you and one Yeah Ready.

So according to me I feel that certain indicators that are required to attract any country for international business for the first is the market size and the penetrate uh huh the growth because as you see the international market There are four types of people that are there one either are the global citizens or their global dreamers or their anti global anti global's they're anti global's or they can they can be global agnostics do not. Baseball takes decisions on a bland global active instead the judge a global product by the same criteria that they use for local for us. So these people are known as global agnostics. Okay So the 4th one that is there is global agnostics. They're measuring their global products like there yeah local products itself. Now the site this is the first indicator which I feel the second indicator would be the competitive in mind. That is very important completely when mine made is very important for something to flourish. 3rd is institutional contexts for this for that we have is the cultural distance, administrative, distance, the geographic distance so we can write it together. The cultural, the administrative, the geographic yeah and the economic distance. So according to me I feel that these are the indicators that attract the country for international business. So I hope

This question let us grow the situation for the diagram for the question. So this is the Earth. Okay, and we can make a tunnel through it. So this is our tunnel, which is unrealistic. Okay, so now a ball has been dropped or an object has been brought here from here and it this is the center and this is our another circle. Okay? It's here. So ball is here at any situation and this previous is small art and this radius is capital RT, which is actual radius of Earth. Okay, this complete Earth has must M and see this has mass M dash. Okay MDS Capital Mds. And this ball is moving in downward direction because of attraction. So this has forced F dash here. Okay. And ah uh okay, so now, since the Earth has uniform density then we can determine this mass M. Dash in the term of mass capital M. S. M dish that will be equals to mass M divided by four by three by R E. Q, Manipulated by four by three by small. R. Q. Okay, so from here we get Mds equals two and by R E. Q. Multiplied by are you okay Now the gravitational force on the ball which has master small M can be written as F equals two minus G. M mm. Dash divided by r squared. Okay, so now substituting value of mds here we get minus G. M by R squared. Multiplied by M by R E. Q. Marie claire by are you okay? So from here after solving bigger to minus G mm. Capital mm by R E Q. Multiplied by smaller. Okay. Now this force can be compared with equation of F equals two minus K X. Where X. Is the displacement is small displacement. And this are is again that the small displacement from the center of the earth? Okay, so now uh if this is a small displacement, then this force F. Creates a restoring force which will allow this ball of mass M. Two mm in shh! Okay. So now from the theory of um the time period can be written as to buy. And the root of MBK were cases constant. And after comparing from the above equations two equations, we can write that K is equal to G. Small M. Capital empty, whereby are you? Okay, so now substituting this value of K. Here we get teeth. Was to buy under root of mm divided by K. S. G. Small M. Capital M. Divided by R. E. Q. Okay, so now from here we can write equals two pi and the root of R. E. Q. Divided by G. M. Okay, this embassy cancel out this M. Will cancel out. Okay, this is the mass of the board. Okay? And this is the mass of earth mask of a. Mhm. Okay. And this is the radius of radius of A. So this is the equation we require as further problems. Okay? So we can rewrite it as to buy under root of R E. U. D. Whereby G capital M. Okay. So this is the answer for the question. This is the time period which we require. Okay. No, thank you.

Hi everyone here it is given two speakers separated apart by a distance of white. 690 m. And powered by the source of frequency 6 85 m. Sorry, 6 55 hertz. Mm hmm. The speed of sound is 3 43 m per second. In the first part we have to find the position of observer facing the body speaker so that he can hear the maximum sound. Suppose he's standing at this point which is extra strength. So here it is an observer. Extra steps from here. So we have to find the value of X nearest four hearing. Mhm maximum sum. And we have to find a venue meadow of export hearing released some course part for he maximum summer at the position of observer. Oh two songs must meet this activity. Yes that is the party fence. A week. I am in tune and here I am baby. Yeah 01 and so on. Yeah. Born into London for nearest so. Bart difference between mhm X minus 0.690 minus X. Has got to be born into Orlando. Your very breath will be speed upon frequency. So we will let you will let 3 43 divided by 169 So it is to be 0.5 00 m. Right? So X minus 0.690 plus X is called do find fights you know as you look. Mhm. So as you will get 1595 m now for second part we do of export hearing. Yeah least son that is destructive interference. Mhm must take place for party friends must be and plus half and to land up here and maybe 0123 and so on. So X minus 16 Thank you minus X. Must be Yeah, half and 2.500 So the value of X. You will get 24 7 m. There's a thanks for budget.


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