Final answer:
If the Earth's radius were halved, its new angular velocity would be four times the original to conserve angular momentum. Therefore, a day would be one-fourth its current length, making a day last approximately 6 hours.
Step-by-step explanation:
If the Earth were to shrink to half its radius due to its own gravity, we can explore the implications using the conservation of angular momentum, as the mass of the Earth would remain constant. Assuming the Earth's shape remains a sphere, and there are no external torques, the angular momentum L of the Earth is conserved. Angular momentum is given by L = Iω, where I is the moment of inertia and ω is the angular velocity.
For a sphere, the moment of inertia is I = ¾MR², where M is the mass and R is the radius. When the radius is halved, the new moment of inertia I' is I' = ¾MR'² = ¾M(½R)² = ¾M¼R². This is one-fourth of the original moment of inertia.
Since L = L', we have Iω = I'ω', where ω is the original angular velocity and ω' is the new angular velocity after the radius has shrunk. Therefore, with I' being one-fourth of I, the new angular velocity ω' must be four times ω to conserve angular momentum. As a result, one day will be four times shorter.
The current length of one solar day is about 24 hours, so if one day became four times shorter, it would be 6 hours long. This is a simplistic model and doesn't take into account other factors that might affect rotation, but it serves as a good approximation for this hypothetical scenario.