Keeping the mass of Earth constant, if the radius is halved, the duration of a day would be shortened. Let's explore the reasons behind this phenomenon.



The duration of a day on Earth is determined by its rotational period, which is the time it takes for the Earth to complete one full rotation on its axis. As the Earth rotates, different parts of the planet experience alternating periods of daylight and darkness.



When the radius of the Earth is halved, it means that the distance from the center of the Earth to any point on its surface is reduced by a factor of two. This reduction in radius has significant implications for the rotation of the Earth.



The moment of inertia of a rotating object depends on its mass distribution and the axis of rotation. Halving the radius of the Earth results in a decrease in the moment of inertia. This is because the mass of the Earth remains constant, but the mass distribution is concentrated closer to the center.



According to the law of conservation of angular momentum, the total angular momentum of a system remains constant unless acted upon by an external torque. Angular momentum is the product of the moment of inertia and the angular velocity.

Since the moment of inertia decreases when the radius is halved, the angular velocity must increase in order to conserve angular momentum. In other words, the Earth would need to rotate faster to maintain its total angular momentum.



As the Earth rotates faster, the duration of a day is shortened. This means that each rotation on its axis takes less time to complete. The exact calculation of the new day duration would depend on the specific values of the original radius and the new radius.

However, it is important to note that altering the Earth's radius in reality would have various other effects on the planet's dynamics, including changes in gravity, atmospheric conditions, and tides. Therefore, this theoretical scenario assumes that all other factors remain constant except for the radius.

In conclusion, if the mass of the Earth is kept constant while halving its radius, the duration of a day would be shortened due to the conservation of angular momentum and the need for the Earth to rotate faster to maintain its total angular momentum.

T*2 directly proportional to R*3