Question

1. The angular displacement e(t) of an undamped pendulum of length l swinging in a vertical plane under the influence of gravity can be modelled with the nonlinear second order ODE 0"(t) + "$(0(e) – Pef) = = 0, (1) where w2 = { (a) Re-write equation (1) as a system of first-order ODEs by defining (1 = 0(t) and 02 = 't). (b) Find the nullclines and equilibrium points for the system of first-order ODEs. (c) The first order ODE that describes trajectories in the (0,02) phase plane is given by dezwa (-60) de 602 Re-write the ODE in differential form and use the solution method for exact equations to find the general solution. You MUST clearly show ALL the steps in the exact solution method as detailed in the lecture notes and lecture recordings. (d) Determine the equation of the phase trajectory that satisfies the conditions 0(0) == 8 and 0 (0) = 0. (e) Sketch the trajectory in the (0.02) phase plane that satisfies the conditions y(0) B, with 0 < 8 < 76/2, and /(0- 0. Clearly label your sketch with: (i) the intersection points of the mullelines with the phase trajectory and (ii) the equilibrium points of the system, (f) Write down the values of (0,0%) that correspond to the maximum value of the gravitational potential energy for the motion of the pendulum satisfying the initial conditions 0(0) B. with 08/2, and 0 (0) = 0. (g) Write down the values of (0,0%) that correspond to the maximum value of the kinetic energy for the motion of the pendulum satisfying the initial conditions 0(0) = 3, with 0 < 3 < 70/2, and (0) == 0.

Answer 1

la) 0 Let diz o 4 d = 6, So d få -W. (0,-./6.) the linear system This ใน (b) 0,70 =) On or = = oi 03/6=0 o. -> 0,=0 4 or ab. b O, do 20 fo ht to nale ling

7 m = DO ( te) M Clusi) dont N. (@1,80.dll; = 0, Where = an (02-60) =-6.bg a @N= 20 Jo the equation is exact. (mdor = 30 -4 In de 304 The solution 30, = e. d). or oß po gives, WHAB 38) The solution trajectory i Ay, –321) 324) 3.89 W 2001 - 30 14 Og =C az A W A 312)