4. Consider a series of two springs with spring constants kj and 2, respectively, connected in series with interconnected masses m, and m2 as shown in the diagram below. Hooke's law states that a spring extended a length L experiences a contraction force equal to kL where k is the spring constant. (a) The two masses are located at distances x1 and 22 away from a fixed point from which the springs hang under gravity. By considering the balance of forces on each of the two masses, justify, perhaps with a force diagram or an explanation, why , and 72 must satisfy mix1 = -kid1 + mig + k2(12 - 21), (3) m212 = -k2(12 - 21) + m2g. k m1 K2 m2 (b) Eliminate x1 from the system of equations above to show that 12 must satisfy the fourth-order linear, non-homogeneous differential equation mim212+ (mik2 + maki + mzk2) x2 + (kik2)12 = g (maki + (mi + mz)k2). (4 ) (c) From now on set m1 = m2 = 1. Show that the roots to the characteristic equation for (4) are purely imaginary numbers tow, and tow2 which you must determine in terms of kj and k2. Hence, find the general solution to (4). (d) Optional: Consider a mass m attached to a single spring of spring constant k hanging under gravity as shown below. Find a differential equation for x and compare this to the equation obtained by taking the limit in (4) as mi - 0. Hence, show that the addition of two springs of spring constant ki and k2 attached in series behaves as a single spring whose spring constant is the harmonic mean k = 7 K1k2 K1 + k2 k m