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I've come to understand that the torque acting on the Earth while revolving the Earth is zero. Torque is the force responsible for rotation of a body. So how does the Earth rotate?

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Moving bodies have inertia which means that they will continue to move at a constant velocity unless acted upon by an external force (this is Newton's first law of motion).

Similarly, rotating bodies have a moment of inertia, meaning that they will continue to rotate unless acted upon by an external force (torque).

Therefore, torque is only required to either accelerate (or decelerate) a rotating body, or to act against a resistive force so as to maintain a constant rotational velocity.

Since there is only a very tiny resistance to the rotation of the earth, no torque is required. Note, however, that since there is resistance (primarily from tidal forces) then the earth is actually slowing down.

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