Why a swinging pendulum does not rise to exactly the same height it initially started from on the other side?

Because it does work on the surrounding air

Explanation

A pendulum will always rise to the height from which it fell, unless it works on something on its way, e.g. touching an object while swinging.  if a pendulum does not hit anything as it swings, it rises very nearly to the same height as that from which it was released.  The reason that it does not quite reach the height from which it was released is that the pendulum doe work on something as it moves-it does work on the surrounding air.  The pendulum exerts a force on air molecules and pushes them aside.  But if a pendulum is not moving too fast, it does not exert a great amount of force on the air molecules, and it rises very nearly to its original height.