The restoring force of a simple pendulum in the angular (6) direction is F₂ = -mg sin 0, and the corresponding acceleration is a the angular acceleration == -Fe/m = -g sin 0. Dividing ao by the pendulum length L yields g απ = a/L sin L ΦΩ We know from rotational kinematics that angular acceleration is the rate of change of angular velocity, i.e. a = $, and the angular velocity is the rate of change of the angle ǝ, i.e. dt Ω de dt Therefore, our differential equations of motion for the simple pendulum are ΦΩ g sin dt de Ω dt Why do we use a capital greek omega (n) for the angular velocity? Because the lowercase wis defined in the lab manual as the angular frequency of the pendulum, which is a different (constant) quantity: g Iteration is King (again)! The Euler-Cromer method of numerical integration To solve these differential equations, we use (almost) the same iterative Euler method as for the "free fall/air drag" experiment. First, let's define the initial values: We start the pendulum at time to from an angle 00 with zero angular velocity (0 = 0). For a sufficiently short time interval At, the angular acceleration can be considered constant, so we can calculate the new angular velocity at time t₁ = to + At: Ω1 = Ωο + ΦΩ dt At: = No - - sin (00) At Likewise, the angular velocity does not change much during At, so the new angle at time t₁ is 01 = θο + Ω1Δt We repeat these calculations over and over. For the ith step, we have ti = ti−1 + At N₁ = n;-1 - 1-sin(04-1) At L θ; = 0-1 + Ω;ΔΙ 21 Write the code for calculating tį, Si and Oi, based on the equations above. As before, we use a for loop for the repetitive (iterative) calculations. Remember that only indented lines become part of the loop, and all those lines must have the same indentation. • The sin function comes from the numpy library, so call it as np.sin(). • For the time step, use the variable dt defined earlier. • The i-th elements of time, angle and angular velocity can be accessed as t[i], theta[i] and Omega [i], respectively. For element number i - 1, replace i with i-1. [ ] : # Initial conditions. Vector numbering in Python starts at 0. time [0] = = 0 theta [0] = Omega [0] = 0 # initial time np.deg2rad(5.0) # initial angle, converted to radians. # initial angular velocity (pendulum starts at rest) # Main loop: Euler algorithm and evaluation of exact solutions for v and y # # Your code should look like this (without the # signs, of course): # for i in range(1, t_steps): # time[i] Omega[i] = .... = .... # theta[i] = # YOUR CODE HERE raise NotImplementedError()