Problem set 4Ge 108November 30, 20121 Double your pleasureConsider a system, as illustrated below, that has two pendula of length l, witha mass m at the bottom of each, that are connected by a spring with springconstant k.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEv v} }m ml lθ1θ2¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢AAAAAAAAAAAAAAAAAAAAk(a) What are the equations of motion for the two masses, assuming thatthe pendulum motions are small (so θ = sin θ) and that the spring can beapproximated to be completely horizonal at all times (this would be impossible,of course, but what we are really assuming is that the vertical motions of thesprings are much much smaller than the length of the spring, so that we needonly consider the horizonal motion of the masses for calculating the stretchingof the spring).1(b) What are the frequencies of this oscillation?(c) What are the modes?2 L.A. StringsFor an infinite string with mass per unit length p and tension T , we found thewave equation to be∂2u∂t2= v2∂2u∂x2and we found that one solution to the equation wasu(x, t) = u1exp(ikx + ikvt) + u2exp(ikx − ikvt).Show that, in fact, any function of the formu(x, t) = f (x + vt) + g(x − vt)is a solution to the wave equation.Consider now a function h(x) where, for all values of x less than zero orgreater than one, the function is equal to zero. Between 0 and 1, though h(x)is a silouhette of the downtown LA skyline (hmmm... is there such a thing?).If u(0) = h(x) and ˙u(0) = vdhdx, what does the solution to the wave equationlook like at a later time? Sketch the solution at a few times in the future. Whatif ˙u(0) = −vdhdx? If ˙u(0) = 0?3 Seriously coupled springsProgram springs.m calculates the position of a set of n masses connected byn + 1 springs which are eventually connected to the wall. As the programis currently set up, the number of masses is 19, the number of springs is 20,all masses are the same, all springs are the same, and the system is initiallyperturbed by moving the first mass away from the wall.The program works by calculating the acceleration felt by each mass, firstfrom the spring on the left, then from the spring on the right, and keeping trackof the changing velocity and position of the mass.(a) Figure out how the program works (or, alternatively, if you prefer, writeyour own program from scratch).(b) Run the program. The initial perturbation eventually travels all the wayto the right wall. How long does it take for this signal to reach the right wall?Modify the program so that all of the spring constants are 10 times higher. Howlong does the signal take to reach the right wall now? What do you think thevelocity of signal propagation is, in terms of k, m, and l (just use m and l tomake the units come out correctly).2(c) Modify the program so that the motions are now damped. Use a dampingconstant of 1, in the units of the problem. How long does it take for most ofthe motion to die away? What if the damping constant is