The spring inside BC is of constant k1 = 1000 N/m. The spring of AA0 is attached to the collar C and has a constant k2 = 800N/m. Both springs are unstreteched at the shown configuration. The collar is of mass 10 Kg. Assume that the rods, shafts & spring are massless, and that the springs can be stretched or compressed as much as necessary. Acceleration due to gravity, g = 9:8 m/s2, acts downwards.
1. Write down the velocity of collar C in terms of the variables (r1; _1) and their time derivatives. Similarly, write down the velocity of collar C in terms of the variables (r2; _2) and their time derivatives.
2. Write down the constraint equation(s) that impose the constraint that the collar, C, is attached to the end of BC and is constrained to slide along the shaft of AA0. Write your answer in terms of the derivatives of the chosen coordinates (r1; _1; r2; _2). Is/are these constraints holonomic or non-holonomic? Explain your answer.
3. Find the equilibrium con_guration(s) of the system. Identify the stable and unstable equilibria.
4. Write down the equations of motion of the collar in terms of the chosen generalized coordinates and their second time derivatives.
5. Write the constraint equation(s) and the equation(s) of motion as a set of _rst order o.d.e.'s (involving the variables (r1; _1; r2; _2)). Write these equations in form of a matrix equation.
6. Write a MATLAB/Octave program to numerically integrate the above set of o.d.e.'s. Integrate the equations with the initial condition that the system is at rest in the shown configuration. Plot the results (plot r1; _1; r2; _2 as a function of t for t = 0 s to t = 10 s).
7. Using the above numerical integration result, create a simple simulation/animation in MATLAB/Octave of the system by plotting the line segments AC and BC at the different time instants.
