Two uniform slender rods of identical length \(2l\) and identical mass \(m\) are interconnected by a smooth pin at \(A\). Rod 1 is connected to the fixed support 0 by a smooth pin at \(O\). The end \(B\) of rod 2 slides along a vertical frictionless slot. The mass centers \(G_1\) and \(G_2\) are connected by a spring of constant \(k\). The system is released at time \(t_0\) from rest from the position shown, the spring being unstretched. See Figure 1 (left) below.
The goal of this question is to find the dynamics of the system immediately before point \(A\) reaches its lowest point (time \(t=t_1\)): see Figure 1 (right).
1. (10 pts) Find the location of instantaneous center \(I_2\) of rotation of rod 2 at time \(t_1\). From the knowledge of this point, find kinematical properties of the system at time \(t_1\).
2. (20 pts) Now apply the Work-Energy Principle between \(t_0\) and \(t_1\) to find the speed of point \(B\) at time \(t_1\).
Answer: \(v_B^2 = 12 \sqrt{2} gl\) (to be verified!).
3. (20 pts) Generalize the results of question 2: find the speed of \(B\) at any time \(t\), for \(t_0 \leq t \leq t_1\), by first parametrizing the position of the system. Is the application of the WEP sufficient to predict the evolution of the system?
Note 1: The moment of inertia of a slender rod of mass \(m\), of length \(l\), of mass center \(G\) about axis \(Gz\) is \(\frac{1}{12}ml^2\).
Note 2: Do not replace the initial and final angles displayed in the Figure by symbols.