I'm trying to derive the periodic time of a pendulum motion. I have started with writing the height of the object as a function of the angle . Then I have calculated . After that I have written the gravitational potential energy as , so I got an expression for as a function of , but since I'm only working with small angles I have set the second derivative to a constant, so . Then I have written an expression for the kinetic energy as a proportion of the angular velocity in square, .
So, I have:
Since the total amount of energy, is always the same, I obtained , so I got the new system
and by substitution I could remove as well from the system, so I could minimize it to a single equation:
Now, this is where I got stuck. I don't know how to make it from here. Am I on the right track, and in that case can someone explain to me how to continue? Besides, can I make the assumption that for all alpha, since I am only woring with small angles? Thanks in advance!