How long do the Earth and the Moon take complete one orbit around their barycentre?

27-1/3 days. If the moon’s position is observed in the sky with reference to any star (not planet), the Moon will return to the same position in the sky (with reference to that star) after a period of 27-1/3 days.  

Why does the Moon rise later each night by an average of 50 minutes?

The period of the Moon’s revolution around the Earth is not an extract multiple of the period of rotation of the Earth around its axis.






What do we mean by “Sidereal Period”, “Synodic Month” and “Lunation” with reference to the Moon?

The period of 27-1/3 days for the Moon’s orbit is known as the “Sidereal Period” or “Sidereal Month”.

The period of 29-1/2 days for the Moon’s phases is known as a “Synodic Month” or “Lunation”. 







What is there a difference between the sidereal period and the synodic month (Lunation)?

The answer lies in the changes in the relative positions of the Earth, Moon and Sun during the period of one orbit of the Moon. It must be remembered that while the Moon is orbiting the Earth, both are also orbiting the Sun.









What is the significance of the “Sidereal Period” of the Moon?

In one “Sidereal Period” the Moon completes one orbit of the Earth measured relative to the background stars which, for this purpose, are considered fixed in position.