homework and exercises - Why do we experience the normal force in an elevator and not the net force?

In a book I saw that the time period of a pendulum inside a elevator moving up is, $$T=2\pi\sqrt{\frac{L}{g+a}}$$ I was curious as to why we use $(g+a)$ as we know inside an elevator,

$F_{net}=F_n-mg$

or,$ma_{net}=F_n-mg$

or,$F_n=m(a_{net}+g)$

So my question is shouldn't we use $a_{net}$ instead of the acceleration due to the normal force?

Answer

The net force on the pendulum is:

$${\vec F}_{net}=-m{\vec g} -{\vec F}_n$$

so

$$m{\vec a}_{net}=-m{\vec g}-m{\vec a}_n$$

dividing both sides by $m$:

$${\vec a}_{net}= -{\vec g}-\vec a_n$$

where both $\vec g$ and $\vec a_n$ point downwards. You state instead that (and I think that this is the source of confusion) $\vec F_n$, and thus $\vec a_n$ points upwards, which is the case for the lift, but the normal reaction force (which the pendulum experiences), which is equal in magnitude to the acceleration of the lift $\vec a$ is pointing downwards. So the magnitude of the net acceleration is $g+a_n$, pointing downwards, which we obviously can't use as $a_n$.