Problem 3-Conservation of angular momentum and physical pendulum A system is composed of a thin uniform rod of length L= 1 m and mass M = 1 kg, and a bullet of mass mb - 7 g. Initial, the rod is in equilibrium against the force of gravity, with its axis along the vertical direction, pivoted at a distance L/4 from the rod's upper end; the bullet travels along the x-direction, with initial velocity v = 100 m/s, until it collides against the rod at a distance L/4 from the rod's lower end. After the collision, the bullet emerges from the rod with a velocity v' = 50 m/s along the x-direction, and the rod starts rotating around its pivot point. (a) Find the initial angular velocity of the rod. (b) Use conservation of energy to find the maximum angle reached by rod. (c) Find the time it takes for the rod to return to its original position, in the limit where the maximum angle reached by the rod is small (Hint: in part b, use 1 - cos a = 2 (sin a/2)² ).

I attached the question and the answer What I don’t understand in part a is the part after we find angular velocity how was the semplification made to find the maximum angle just the simplification And in part b I don’t understand where pi/4 comes from

Please solve #5

a shower if JU equivalent situation, a rolle FIGURE 8.14 Shows a roller-coaster car gu radius r. Why doesn't the car fall off at the top of the circle eircle is not uniform circular motion; the car slows down as it goes up one and speeds up as it comes back down the other. But at the very top tom points, only the car's direction is changing, not its speed, so at those points the acceleration is purely centripetal. Thus there must be a net force toward the center The aus poats toward abe center and very bot- motion, so the water trajectory toward yo of the circle. 8-8 CHAPTER 8 Dynamics II: Motion in a Plane 15. It's been proposed that future space stations create "artificial gravity" by rotating around an axis. a. How would this work? Explain. b. Would the artificial gravity be equally effective throughout the space station? If not, where in space station would the residents want to live and work?