I am wondering how the Earth got its original angular velocity. What was the cause that made the Earth rotate about its own axis?
What about its motion around the Sun? What gave the Earth the push to keep rotating around the Sun?
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Origin of Earth rotation
There are two main phenomena that explain Earth rotation. First one is linked to the history of the solar system: there is angular momentum in the interstellar medium (ISM), so the collapsing cloud that formed our solar system had an initial angular momentum (rotation of the galaxy and turbulence in the ISM are typically a good candidate for that; see for example Mac Low & Klessen 2004). By conservation of the latter, particles in the collapsing cloud accelerate (the norm of the angular momentum $I$ is $mrv$, with $m$ the mass of the system, $r$ the radius and $v$ the velocity; as the cloud contracts, $r$ decreases and then $v$ has to increase) and due to centrifugal force, a disk will tend to form. By conservation of the angular momentum in the protoplanetary disk, you will also tend to form bodies that rotates.
The second phenomena is linked to the formation history of the Earth. Simply put, Earth formation history is due to collisions of planetesimals (let say they are big chunks of rock, formed by a collision history of smaller chunks of rock, formed by...); these planetesimals are orbiting the forming star in the protoplanetary disk, their rotation being linked to the disk formation history (see above). When planetesimals collide, two things can basically happen: either the collision is perfectly central (meaning that the momentum of one of the two colliding objects is pointed toward the center of mass of the second object) and there is no transfer of angular momentum, either the collision is lateral and there is a transfer of angular momentum. Depending on the configuration, you can either transfer momentum that will make the other body rotating on way or the other. That being said, you can then object that there should be a equirepartition of collisions pushing the proto-Earth to rotate in both directions. Which is true, but external planetesimals are rotating faster than internal ones. Therefore, from the proto-Earth perspective (meaning, in a physical sense, in its reference frame), internal planetesimals are seen as coming backward, from front, and external planetesimals are seen as coming forward, from behind. Long story short, there are both transfering momentum in the same "direction", forcing the proto-Earth to rotate in the same direction as the rotation of the protoplanetary disk (see for example Artem'ev & Radzievskii 1965).
Rotation around the Sun
This one's easy: gravity.
The Earth is constantly falling towards the Sun because of gravity, and the centripetal acceleration counteracts perfectly gravity, mainting the Earth on its orbit.
