Question

If the earth shrinks until its radius becomes \(\frac{3}{4}\) of its original, the new duration of the day would be (mass unchanged):

• A. \(24 { hours}\)
• B. \(16 { hours}\)
• C. \(13.5 { hours}\)
• D. \(18.5 { hours}\)

Hint:

Remember that when a celestial body's radius changes, its moment of inertia and thus the duration of its rotational period also changes inversely with the square of the radius.

Answer 1

The Correct Option is C

The duration of the day on Earth is related to the moment of inertia and the angular velocity of the Earth. When the radius of the Earth shrinks, the moment of inertia decreases, causing the Earth to spin faster in order to conserve angular momentum. We can apply the principle of conservation of angular momentum, which states that: \[ I_1 \omega_1 = I_2 \omega_2 \] where: - \( I_1 \) and \( I_2 \) are the moments of inertia at the initial and final states, - \( \omega_1 \) and \( \omega_2 \) are the angular velocities at the initial and final states. The moment of inertia for a sphere is given by: \[ I = k m R^2 \] where: - \( m \) is the mass of the Earth, - \( R \) is the radius of the Earth, - \( k \) is a constant depending on the distribution of mass (which doesn't change in this case). Since the mass of the Earth remains unchanged, we can ignore \( m \) and \( k \) in the equation. The angular velocity \( \omega \) is inversely proportional to the moment of inertia, so: \[ \omega_1 R_1^2 = \omega_2 R_2^2 \] Substitute the given values: - \( R_2 = \frac{3}{4} R_1 \), - We need to find the new duration of the day, which is related to the angular velocity by \( T = \frac{2\pi}{\omega} \). Thus, we have: \[ \frac{\omega_2}{\omega_1} = \frac{R_1^2}{R_2^2} = \frac{R_1^2}{\left(\frac{3}{4} R_1\right)^2} = \frac{1}{\left(\frac{3}{4}\right)^2} = \frac{16}{9} \] The new angular velocity is \( \omega_2 = \frac{16}{9} \omega_1 \). Since the period \( T \) is inversely proportional to the angular velocity, the new period \( T_2 \) will be: \[ T_2 = T_1 \times \frac{9}{16} \] The original period \( T_1 = 24 \) hours, so: \[ T_2 = 24 \times \frac{9}{16} = 13.5 \, {hours} \] Conclusion: The new duration of the day is 13.5 hours, so the correct answer is (3).