Question

The Problem The single pendulum Consider the single pendulum shown below. There is a bob at the end of the pendulum, of mass T32 For soscillations in the case of a frictionless pivot and a rod with negligible mass, the motion of the pendulv time can be described by the second-order linear ODE where 0 is the angle between the rod and the vertical, g is gravity, E is the length of the rod Q1 By substituting (2) into the ODE, terify that the general solution to equation (1) is given by where w Vg/e. The double pendulum A double pendulum consists of two pendulums attached end-to-end, as shown below.

Answer 1

"Theta = -g theta/l = > d^2 theta/dt^2 + g/l theta = 0 = > d^2 theta/dt^2 + omega^2 theta = 0 [omega^2 = g/l] Substituting theta(t) = C_1 cos(omega t) + C_2 sin(omega t) = > -C_1 omega^2 cos(omega t) + -C_2 omega^2 sin(omega t) + omega^2 C_1 cos omega t + C_2 omega^2 sin omega t = 0 Hence theta(t) = C_1 cos(omega t) + C_2 sin(omega t) is a solution of theta = -q/l theta Double pendulum theta_1 = -2 omega^2 theta_1 + omega^2 theta_2 theta_2 = 2 omega^2 theta_1 - 2 omega^2 theta_2 omega^2 = g/l l = 10 m g = 9.8 ms^-2 phi"" = -omega^2 A phi phi = [theta_1 theta_2] = -omega^2[2 -2 -1 2][theta_1 theta_2] A = [2 -2 -1 2]"

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