Question

Because the Earth is nearly perfectly spherical, gravitational forces act on it effectively through its center. Explain why the Earth’s axis points at the North star all year long. Also explain why the earth’s rotation speed stays the same throughout the year (one rotation per $\mathbf{24}\mathit{h}$). In your analysis, does it matter that the Earth is going around the sun?

In actual fact, the Earth is not perfectly spherical. It bulges out a bit at the equator, and tides tend to pile up water at one side of the ocean. As a result, there are small torque exerted on the Earth by other bodies, mainly the sun and the moon. Over many thousands of years there are changes in what portion of sky the Earth’s axis points towards (Change of direction of rotational angular momentum), and changes in the length of a day (change of magnitude of rotational angular momentum).

Short answer

The rotating angular momentum of the Earth-sun system has no effect on the rotational angular momentum of the earth's center of mass. As a result, the fact that the Earth revolves around the sun is irrelevant.

Hence, it doesn’t matter that Earth is going around the sun.

Step-by-step solution

Given Data

Gravitational forces act on the Earth effectively through its core because it is nearly perfectly spherical.

The Earth is not exactly round in reality. The equator bulges a little, and tides tend to pile up water on one side of the ocean. As a result, other bodies, primarily the sun and the moon, exert minor torques on the Earth. The location of the Earth's axis in the sky has changed throughout thousands of years.

Concept of Angular Momentum

Angular momentum is a characteristic that describes the rotating inertia of an object or set of objects moving around an axis that may or may not pass through it.

Determine where the Earth's axis points to the North and the earth's rotation speed remains constant throughout the year

As the earth is nearly perfectly spherical, the gravitational force acts on it effectively through its center, which means the gravitational force and the center of mass act at the same point then the torque acting about the center of mass is zero. Thus, the corresponding rotational angular momentum stays constant in magnitude and direction (the direction of the rotational angular momentum indicates that the Earth axis points at the North Star all over the year, and the magnitude of the rotational angular momentum indicates that the Earth's rotational speed stays the same throughout the year). The rotational angular momentum relates to center of mass of the earth is not affected by the rotational angular momentum of the Earth-Sun system. Hence, it doesn't matter that Earth is going around the sun.

No torques around the center of mass, so rotational angular momentum stays constant in magnitude and direction, hence, it doesn't matter that Earth is going around the sun.