Correct Answer - Option 1 : 6 Hours
Concept:
Conservation of angular momentum:
A rigid spinning object continues to spin at a constant rate and with a fixed orientation unless influenced by the application of external torque.
i.e., \(\tau = \frac{{dL}}{{dt}}\) and if momentum is conserved there would be no external torque acting on it.
\(\tau = 0 = \frac{{dL}}{{dt}}\;or\;\frac{{{\rm{\Delta }}L}}{{{\rm{\Delta }}t}}\)
- This means that the change in angular momentum is zero if there is no external torque i.e., ΔL = 0. Thus, for a closed system, angular momentum is constant
The angular momentum is given as
L = I ω
I is a moment of inertia, ω is angular velocity.
Calculation:
The moment of inertia of Earth is
\(I = \frac{2}{5} MR^2\)
M is mass, R is the radius
Angular speed is given as
\(ω = \frac{2 \pi }{T}\)
In process of the sinking of the earth, no external torque is applied.
Mass is same, radius changed
So,
I1 ω 1 = I2 ω 2
\(\implies \frac{2}{5} MR^2 (\frac{2 \pi }{T}) = \frac{2}{5} MR'^2 (\frac{2 \pi }{T'})\)
\(\implies (\frac{R^2 }{T}) = (\frac{R'^2 }{T'})\)
The new radius R' = R / 2
New Time Period T' , which we have to find
So,
\(\implies (\frac{R^2 }{T}) = (\frac{(R'/2)^2 }{T'})\)
\(\implies T' = \frac{T}{4}\)
T is 24 hr in normal conditon.
So, T' = 24 hr / 4 = 6 hr
So, the correct option is 6 hr.