homework and exercises - How to derive the period of spring pendulum?

So I wanted to find out how to (simply, if that's possible) derive the formula for a period of spring pendulum: $T=2\pi \sqrt{\frac{m}{k}}$. However, Google doesn't help me here as all I see is the ready-to-bake formula. Could you please point me some directions?

Answer

You need to know the equation of motion. The force for the pendulum is given by $F= - k x$. Newtons equation tell you $F=ma = m \ddot x$. So you need to solve

$$\tag{1} m \ddot x = - k x.$$

You know that the solution will be of oscillatory form. So you set $x= A \cos(2\pi t/T)$ and you want to obtain $T$. Plugging this ansatz into the equation (1), you obtain

$$- m\frac{(2\pi)^2}{T^2} A \cos(2\pi t/T) = - k A \cos(2\pi t/T).$$ You see that the equation is fulfilled if $$m\frac{(2\pi)^2}{T^2} = k.$$ Solving for $T$, you obtain the result.