MATLAB Project - Elbow

Calculations by hand (not by MATLAB) of special values of the angles θ and φ, and when these values occur. By closely watching the simulation, you will see that the maximum value of φ(t) is ¼ Π, and that this value occurs when t = 0, 1, 2, 3, 4, . . . sec.

(a) Draw a sketch of the layout when φ(t) is a minimum. Demonstrate that the minimum value of φ(t) is arccos(7/8) ≈ 0:5054, and give the first time that it occurs.

(b) With the aid of diagrams, explain why the minimum value of θ(t) is arcsin 0:6 ≈ 0:6435 and the maximum value is Π + arctan(15/8) ≈ 4:2224. What are the first times that these values occur?

(c) What is the first time the elbow forms a right-angle, meaning φ = Π/2? What is the second time that P lies directly below A, meaning µ = Π? Give your answers in seconds, to four decimal places.