The duration of the day would be 1.5 hours if the earth suddenly shrinks to 1/64 of its original volume, with its mass remaining unchanged.
The Earth's angular momentum (L) depends on its moment of inertia (I) and its angular velocity (ω):
I = (2/5) * M * R²
L = I * ω
The mass (M) remains unchanged, even though the radius shrinks by a factor of 4 (1/64 = 1/4³), the moment of inertia (I) decreases by a factor of 64 (R³). n order to conserve angular momentum, the angular velocity must increase proportionally to compensate for the decrease in I. Therefore, ω increases by a factor of 64, meaning the Earth spins four times faster.
The duration of a day (T) is inversely proportional to the angular velocity (ω): T = 2π / ω
As ω increases by a factor of 4, the day length will decrease by a factor of 4, becoming 1/4 of its original length. In conclusion, if the Earth's original day length was 24 hours, it would be reduced to 24 hours / 4 = 6 hours after shrinking.