If the earth contracts suddenly, no external torque acts on it, so its angular momentum remains conserved ($L = I\omega = \text{constant}$). The moment of inertia of the earth (assuming it to be a uniform solid sphere) is $I = \frac{2}{5}MR^2$, and its angular velocity is $\omega = \frac{2\pi}{T}$. Equating initial and final angular momenta: $I_1\omega_1 = I_2\omega_2$ $\left(\frac{2}{5}MR_1^2\right)\left(\frac{2\pi}{T_1}\right) = \left(\frac{2}{5}MR_2^2\right)\left(\frac{2\pi}{T_2}\right)$ $\frac{R_1^2}{T_1} = \frac{R_2^2}{T_2}$ Given $R_2 = \frac{R_1}{2}$ and initial time period $T_1 = 24 \text{ hours}$. $T_2 = T_1 \left(\frac{R_2}{R_1}\right)^2 = 24 \times \left(\frac{1}{2}\right)^2 = \frac{24}{4} = 6 \text{ hours}$.