Imagine that the earth suddenly contracts to one-third of its present radius, without any external torque on it. By what duration the day on the earth will decrease considering the earth to be a perfect solid sphere.

Explanation

To solve the problem of how the duration of a day on Earth changes if the Earth contracts to one-third of its present radius, we can follow these steps:

### Step 1: Understand the Initial Conditions
Initially, the Earth has a radius \( R \) and rotates with an angular velocity \( \omega \). The time period \( T \) for one complete rotation (one day) is given as 24 hours.

### Step 2: Calculate the Initial Moment of Inertia
The moment of inertia \( I \) of a solid sphere is given by the formula:
\[
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