Solution

The distribution of the mass of a body is described by the mass moment of inertia J . The mass moves away from the poles, thus away from the axis of rotation, towards the equator. Therefore the mass moment of inertia J increases.

The angular momentum is being conserved as long as there is no impact of external forces. Friction influences are not considered.

The angular momentum L can be calculated from the angular velocity ω and the moment of inertia J in terms of the axis of rotation.

L	angular momentum
J	moment of inertia
ω	angular velocity

Due to the fact that the angular momentum L is being conserved and the mass moment of inertia J increases now, the angular velocity ω must decrease.

A smaller angular velocity means that the earth takes a longer period of time to turn around itself once.

Thus Marie does not have to worry about her sleep at night. If the effect ever had a noticable influence, the nights would be longer and not shorter.