2. A pendulum is a mechanical system in 1which a mass m is attached to a massless inextensible string of length 8 which is in turn connected to a frictionless pivot, as shown at right. The swing angle 6(t) of the pendulum satises the second-order nonlinear differential equation def? 9 . The transformation v = allows us to transform this equation into the separable rst-order differential equa- tion ale 9. _ "E + ?sm6' 0. (l) (a) If the mass m is initally at rest at an angle of 90 = 1r/4 degrees1 solve the differ- ential equation (1) to find 1: = 11(6). (b) Given that dt 1 E = ma (2) integrate both sides of this separable differential equation to nd the general solution if = t(9) in the form of an integral (with respect to 9). (c) Hence, write down a denite integral for the period T of the pendulum (that is, the time for the mass m to complete one full swing and return to its starting position. (Note: 1.1