To solve this problem, we need to understand how the rotation period of the Earth (which determines the duration of a day) is affected by a change in its radius. Let's go through the steps to find the solution.
Relevant Concept:
The rotational period of the Earth is related to its moment of inertia and its angular momentum. If the Earth's radius decreases, its moment of inertia changes, affecting the rotational period. Since angular momentum is conserved, the product of the moment of inertia and angular velocity remains constant. This principle is similar to a figure skater pulling in their arms to spin faster.
Step-by-step Solution:
Understanding Conservation of Angular Momentum: The angular momentum \( L \) of the Earth is given by:
L = I \omega
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. For a sphere, the moment of inertia \( I \) is:
I = \frac{2}{5} m r^2
where \( m \) is the mass and \( r \) is the radius. The angular momentum is conserved, so:
I_1 \omega_1 = I_2 \omega_2
Current State: Let's denote:
\( I_1 = \frac{2}{5} m r^2 \)
\( \omega_1 = \frac{2\pi}{24 \, \text{hours}} \)
New State (after radius reduction): The new radius \( r_2 \) becomes one-fourth of the original, so:
r_2 = \frac{r}{4}
Thus, new moment of inertia \( I_2 \) is:
I_2 = \frac{2}{5} m \left(\frac{r}{4}\right)^2 = \frac{2}{5} m \frac{r^2}{16}\
Finding New Angular Velocity: Use conservation of angular momentum:
\frac{2}{5} m r^2 \cdot \omega_1 = \frac{2}{5} m \frac{r^2}{16} \cdot \omega_2
Solving for \( \omega_2 \), we get:
\omega_2 = 16 \omega_1
Calculating New Day Duration: The new angular velocity is 16 times the original, meaning the Earth rotates 16 times faster. Thus, the new duration of a day is:
\text{New day duration} = \frac{24\, \text{hours}}{16} = 1.5\, \text{hours} = 13.5 \, \text{hours or 13 hours and 30 minutes}
Conclusion:
The correct answer is: 13 hours and 30 mins.
By understanding the conservation of angular momentum, we deduced that reducing the radius increased the angular velocity and consequently reduced the duration of a day to 13 hours and 30 minutes.
