(Solved): The pendulum oscillation is described by the initial-value problem consisting of the equation of m ...

The pendulum oscillation is described by the initial-value problem consisting of the equation of motion with initial conditions:

Here,

is the length of the pendulum,

acceleration due to gravity,

the angle the pendulum makes with the vertical,

the initial angular displacement, and

the initial angular velocity. a) Transform the second-order initial-value problem into a system of first-order initial-value problems. b) Compute the angular displacement

and angular velocity

using the forward Euler method with step size

. Use

, and

.help please A simple pendulum is idealized by considering a bob mass swinging on an inextensible, massless rod (negligible mass relative to the bob) about a frictionless pivot (Figure 1). Figure 1 Simple pendulum The pendulum oscillation is described by the initial-value problem consisting of the equation of motion with initial conditions: \theta ̈+g/Lsin (\theta )=0, \theta (t_0)=\theta _0, \theta ̇(t_0)=\omega _0 Here, L is the length of the pendulum, g acceleration due to gravity, \theta the angle the pendulum makes with the vertical, \theta _0 the initial angular displacement, and \omega _0 the initial angular velocity. a) Transform the second-order initial-value problem into a system of first-order initial-value problems. b) Compute the angular displacement \theta (0.2) and angular velocity \theta (0.2) using the forward Euler method with step size h=0.1. Use L=0.5 m, g=9.81 m / s^2, t_0=0, \theta _0=\pi /4 r a d, and \omega _0 =0 rad / s.