Hi!

I'm trying to derive the periodic time of a pendulum motion. I have started with writing the height $\displaystyle h$ of the object as a function $\displaystyle h(\alpha)$ of the angle $\displaystyle \alpha$. Then I have calculated $\displaystyle h''(0)$. After that I have written the gravitational potential energy $\displaystyle E_g$ as $\displaystyle E_g = mgh$, so I got an expression for $\displaystyle E_g''(0)$ as a function of $\displaystyle \alpha$, but since I'm only working with small angles I have set the second derivative to a constant, so $\displaystyle E_g''(\alpha) = A$. Then I have written an expression for the kinetic energy $\displaystyle E_k$ as a proportion of the angular velocity in square, $\displaystyle E_k=B\cdot\left(\frac{\delta\alpha}{\delta t}\right)^2$.

So, I have:

$\displaystyle \left\{\begin{array}{rl}

\displaystyle{\frac{\delta^2 E_g}{\delta\alpha^2}} = & A\\

E_k = & B\cdot\left(\displaystyle{\frac{\delta\alpha}{\del ta t}}\right)^2

\end{array}\right.$

Since the total amount of energy, $\displaystyle E_{tot}=E_g+E_k$ is always the same, I obtained $\displaystyle \displaystyle{\frac{\delta^2 E_k}{\delta\alpha^2}} = -\displaystyle{\frac{\delta^2 E_g}{\delta\alpha^2}}$, so I got the new system

$\displaystyle \left\{\begin{array}{rl}

\displaystyle{\frac{\delta^2 E_k}{\delta\alpha^2}} = & -A\\

E_k = & B\cdot\left(\displaystyle{\frac{\delta\alpha}{\del ta t}}\right)^2

\end{array}\right.$

and by substitution I could remove $\displaystyle E_k$ as well from the system, so I could minimize it to a single equation:

$\displaystyle \displaystyle{\frac{\delta^2\left(B\cdot\left(\dis playstyle{\frac

{\delta\alpha}{\delta t}}\right)^2\right)}{\delta\alpha^2}} = -A$

$\displaystyle \Leftrightarrow$

$\displaystyle \displaystyle{\frac{\delta^2\left(\left(\displayst yle{\frac

{\delta\alpha}{\delta t}}\right)^2\right)}{\delta\alpha^2}} = -C$

where $\displaystyle C = \frac{A}{B}$

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 $\displaystyle E_g''(\alpha) = A$ for all alpha, since I am only woring with small angles? Thanks in advance!