A material point of mass \(m=500 \mathrm{~g}\) is suspended from a fixed point \(\mathrm{O}\) by an inextensible wire of length \(L=50 \mathrm{~cm}\). The material point, initially in an equilibrium position, is imparted an initial velocity \(v_{0}\). Calculate:

1. the period of the motion, under the assumption of small oscillations;

2. the minimum value of \(v_{0}\) required to make a rotation of \(\theta_{1}=6^{\circ}\) from the equilibrium position and the time taken by \(m\) to reach \(\theta_{1}\). You now want to get \(m\) to an angle \(\theta_{2}=135^{\circ}\) with respect to the equilibrium position. Determine in this case:

3. the minimum value of \(v_{0}\) required for the wire to remain in tension when it reaches \(\theta_{2}\);

4. with this value of the initial velocity, the centripetal and tangential components of the acceleration of the material point \(m\) immediately after departure from the equilibrium point and when it arrives at the position \(\theta_{2}\).