1. The problem statement, all variables and given/known data If you guys cannot read the problem due to all the latex not pasting properly I will post the link to the website. http://courses.ncsu.edu/py411/lec/001/; then proceed to go to the homework section of my webpage then click on Assignment 6 and the problem is next to I. PY411 — HOMEWORK #6 due October 13th Read: Chapter 6. Problems: 5.5, 5.6, 5.9 I. The playground swing is a bit of a mystery! Simply by swinging your feet at the right time (without directly experiencing an external force), the amplitude of your swinging oscillations can grow with time. This is an example of a parametric oscillator. To understand this phenomenon, we start with an undamped simple penduluum described by the equation: d2θ(t) dt2 + ω2(t)θ(t) = 0, where ω2 = ω2o (1 + f (t)) and ωo = pg L ,with g the acceleration of gravity on earth’s surface, L the penduluum length and f (t) is an oscillating function of time (f (t) << 1). You can think of the oscillations as being due to periodic changes in the length of the penduluum, which you cause by kicking your feet. In order to keep things straightforward, pick a general solution for θ(t) of: θ(t) = A(t)eiωp t + B(t)e−iωp t, and an oscillating term f (t) = fo 2i (ei2ωp t − e−i2ωpt), where ωp is the the angular frequency of the solution we are seeking. These solutions and driving terms are not the most general we can select, but the basic physics is included in this treatment. i) As we did in class, plug in the general solution and collect terms with the same time dependence, to derive the following equations which should be true for all times: d2A(t) dt2 + i2ωp dA(t) dt + ¡ ω2o − ω2 p ¢ A(t) + ω2o fo 2i B(t) = 0, d2B(t) dt2 − i2ωp dB(t) dt + ¡ ω2o − ω2 p ¢ B(t) − ω2o fo 2i A(t) = 0. 1 Here we are making the important approximation that we can ignore “rapidly oscillating terms” with oscillation frequencies of 3ωp. These terms do not produce a cumulative effect (unlike the terms oscillating in phase with our solutions proportional to eiωp t and e−iωpt, which do create such effects). ii) Now we will seek solutions to the above equations where A(t) and B(t) are written: A(t) = r(t) cos αo, B(t) = r(t) sinαo. Once again, these are not the most general solutions (α is, in general, a function of time, but the angle ultimately settles down to an equilibrium value we can call αo). Plug these solutions into the equations above and eliminate the terms proportional to d2r(t) dt2 to derive the time dependence of r(t). iii) Can you actually start swinging fom a point where you are motionless and your angular deflection θ(0) = 0? How does this differ from the standard, driven oscillator? iv) Where does the energy come from to increase the amplitude of your swinging motion? 2. Relevant equations 3. The attempt at a solution Isn't this a driven harmonic system problem since the child is making the swing move back and forth? i) I think my general solution is [tex]\theta[/tex](t)= A(t)eiw(p)t+Be-iw(p)t do I need to take the derrivative of theta twice to arrive at : d^2A(t)/dt^2+iw(p)dA(t)/dt+(w(0)^2-w(p)^2)A(t)+w(0)^2*f(0)/2iB(t)=0 d^2B(t)/dt^2-iw(p)dA(t)/dt+(w(0)^2-w(p)^2)A(t)+w(0)^2*f(0)/2iB(t)=0 since f(t)<<1 should I just assume f(t) is negliglbe and w^2=g/L? ii) For this part, I am just plugging in the first and second derivatives of A(t) and B(t) into the solutions I am supposed to derived in ii right?