Final answer:
The Earth's new day would be 6 hours if it contracted to half its present size without any change in mass, due to conservation of angular momentum and the relationship between moment of inertia and radius.
Step-by-step explanation:
If the Earth were to suddenly contract to half its present size, without any change in its mass, the duration of the new day would decrease because the moment of inertia of the Earth would decrease. According to the conservation of angular momentum, when the radius decreases, the rotational speed must increase to conserve angular momentum (L = Iω, where L is angular momentum, I is moment of inertia, and ω is angular velocity). Therefore, the new period of rotation would be shorter because Earth would spin faster.
To calculate this, we use the relationship between moment of inertia (I) and radius (r) for a sphere: I is proportional to the square of the radius. If the radius is halved, the new moment of inertia would be one quarter of the original. Since angular momentum (L) must be conserved, and the mass (m) stays the same, the angular velocity (ω) must increase by a factor of 4 (L = Iω = constant). If the original period (T) was 24 hours, the new period (T') will be T/4, which is 24 hours / 4 = 6 hours.