Question

if tje time period of earth gets doubled without any ecternal torque , then find its new moment of inertia if the initial one is I
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Answer 1

Given:

Initial inertia is I , time period of rotation doubles.

To find:

New moment of inertia ?

Calculation:

$\therefore \: T = \dfrac{2\pi r}{v}=\dfrac{2\pi}{ \omega}$

$\implies T \propto \dfrac{1}{\omega}$

Now, new time period is 2T .

$\implies \: \dfrac{2T}{T} = \dfrac{ \omega}{ \omega_{2}}$

$\implies \: \omega_{2} = </p><p>\dfrac{ \omega}{2}$

Now, since there is no external torque, angular momentum will be conserved.

$\therefore I\omega=constant$

$\implies\: I_{1} \omega_{1} =I_{2} \omega_{2}$

$\implies \: I\omega =I_{2} (\dfrac{\omega}{2})$

$\implies \: I_{2} =2 I$

So, new moment of inertia is 2I

Answer 2

The new moment of inertia of Earth is 2I

Explanation:

Given:

The time period of Earth gets doubled without any external torque

The initial moment of Inertia of Earth is I

To find out:

The new moment of inertia of Earth

Solution:

We know that

If time period if T then the angular velocity

$\omega=\frac{2\pi}{T}$

Let the new moment of inertia be I'

Given that time period gets doubled

Thus, new time period = 2T

new angular velocity

$\omega'=\frac{2\pi}{2T}$

Since there is no external torque

Therefore, the angular momentum will be conserved

From the conservation of anular momentum

$I\omega=I'\omega'$

$\implies I\times\frac{2\pi}{T}=I'\times\frac{2\pi}{2T}$

$\implies I'=2I$

Hope this answer is helpful.

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