In a computer experiment, the main idea of numerical calculations is to divide the trajectory of motion into small sections, in which the motion is described approximately.
We divide the interval of motion from ϕ=ϕ0 to ϕ=0 into N equal intervals of width Δϕ. Let us denote the splitting points as ϕk, k=0,1,...N and the angular velocities at these points as ωk. The main approximation used in further calculations is that at each interval from ϕk to ϕk+1 the motion of the pendulum is considered uniformly accelerated.
For a given amplitude ϕ0 and the number of partition intervals N, answer the following questions:
2.1 Determine the partition interval Δϕ.
2.2 Determine the coordinates of the splitting points ϕk.
2.3 Express the angular velocity ωk at the point ϕk at an arbitrary initial angle of deflection ϕ0. Write down this formula for a particular case of ϕ0=2π.
2.4 Determine the travel time Δtk for the k-th interval from ϕk to ϕk−1.
2.5 Find an expression for the time tk it takes the ball to reach the angle ϕk. To simplify matters, express it in terms of the travel time tk−1 to the previous value of the angle ϕk−1.
2.6 Put down a formula for the oscillation period TN for a given split into intervals.
ATTENTION! Use the dimensionless quantities introduced in problem 1: time τ, period T~ and angular velocity ω~, which are denoted as t, T and ω.