Problem 2 (50 points): Show that the equation of motion for the oscillations of a pendulum (with initial displacement 6y and zero angular velocity) is given by 2 % = % sin 6 (3) where is the gravity constant and [ the pendulum length. (a) Show whether this equation is a linear or non-linear differential equation, i.e., if both #; () and 62 (t) are solutions of the equation above, is 87 (t) + 03 (t) also a solution? (b) Solve Eq. (3) for small oscillations and show the final expression for (). Using this expression identify that the period of the oscillatory movement is given by T = 271'\\/% . () Now, without using any sort of approximation, show that the Eq. (3) can be mapped into do g ?_ \\m 4) If you feel that is too difficult, follow the following steps. 1. First multiply Eq. (3) by df/dt. Now use the chain rule identities 4 (%)2 = 2%% and d sinf x 9 = 4 (cos @), and obtain d dt 2. Integrate the above equation with respect to time. You will obtain a constant. How does the value of this constant relate to do? (d) Find an approximation to the period T, up to second order in 0. If you feel that is too difficult, follow the following steps. 1. Make use of the identity cos 0 = 1 - 2 sin (0/2). 2. Make the change of variables, sin d - sin(0/2) sin(00/2) 3. Use Taylor and expand your equation in powers of 0 and show that 2 T ~ 271 - 1