Understanding the Impact of Doubling Earth’s Radius on Its Rotation Period

Imagine a scenario where the radius of the Earth were doubled while keeping its mass constant. Would the length of a day change? To answer this question, we need to explore how such a scenario would affect the Earth's moment of inertia and, consequently, its rotational velocity.

The Current Angular Velocity of the Earth

The current angular velocity omega;Earth of the Earth is given by the formula:

omega;Earth pi;/TEarth

where TEarth is the length of a day, approximately 86400 seconds.

Effect on Moment of Inertia

The moment of inertia of a solid sphere is given by the formula:

I (2/5)mr2

If the radius r doubles, the new moment of inertia I becomes:

I (2/5)m(2r)2 (2/5)m4r2 (8/5)mr2 4I

Thus, the new moment of inertia is four times the original moment of inertia.

Conservation of Angular Momentum

Since the mass remains constant and no external torques act on the system, angular momentum L is conserved. Angular momentum is given by:

L Iomega;

Therefore, before and after the change, we have:

Iomega; Iomega;

Substituting I 4I:

Iomega; 4Iomega;

This simplifies to:

omega; omega;/4

Thus, the new angular velocity omega; is one-fourth of the original angular velocity.

Calculating the New Length of a Day

If the new angular velocity omega; is one-fourth of the original omega, the new period T can be calculated as:

T 2pi;/omega; 2pi;/(omega;/4) 4 (2pi;/omega;) 4TEarth

Given that the original length of a day TEarth is about 86400 seconds, the new length of a day would be:

86400 seconds x 4 345600 seconds

Converting this into hours:

345600 seconds / 3600 seconds/hour 96 hours

Thus, if the radius of Earth doubled and the mass remained constant, the length of a day would increase to 96 hours.

Quick Question and Discussion

However, the scenario involves several assumptions and the actual implementation is highly theoretical. How would you actually double the mass of the Earth? A collision with a similar sized object, or a hypothetical magic mass-doubling ray gun? This brings up a series of new questions:

Is this about conservation of angular momentum? The answer depends on the method used to double the mass, which introduces further complexities. Can a magic mass-doubling ray gun work? It would likely involve either increasing the size or density or both, contradicting the given parameters. Is the Earth’s mass constant? The length of a day is influenced by rotational velocity, not mass, and thus is unaffected by the metric of mass.

These considerations highlight the intricate nature of theoretical physics and the importance of consistency in physical principles.

Summary

In conclusion, if the radius of Earth doubled while keeping the mass constant, the length of a day would be 96 hours, assuming no external factors come into play. This scenario provides a fascinating glimpse into the interplay of physics principles and their real-world implications.