# Calculating Earth's Rotational Inertia Increase at Tangent to Equator

• lilholtzie
In summary, the conversation discusses the concept of rotational inertia or moment of inertia and how it would change if the Earth's axis of rotation were located at a position tangent to the equator. The parallel axis theorem is mentioned, which states that the moment of inertia about an axis parallel to the original axis but offset by a distance d is equal to the original moment of inertia plus the product of the mass and the square of the offset distance. The conversation also clarifies that the formula for the moment of inertia of a sphere is I=mr^2, but different shaped bodies have different expressions. The conversation ends with a calculation using the given mass and radius to determine the final answer.

#### lilholtzie

1. If the Earth's axis of rotation were located at a position tangent to the equator, how much more rotational intertia (moment of inerti) would the Earth have?

2. I (of sphere) = mr^2

3. I have absolutely no idea where to start.

Look up "Parallel Axis Theorem" in your text.

We don't use our books in class as far as the lessons, so I looked online and is this equation: I=Md^2 the same thing you are talking about?

lilholtzie said:
We don't use our books in class as far as the lessons, so I looked online and is this equation: I=Md^2 the same thing you are talking about?

That looks incomplete. It should include the original centroidal moment of inertia (about a principle axis).

If Ic is the original moment of inertia about some axis, then the moment of inertia about an axis parallel to the original axis but offset by distance d would be I = Ic + Md2, where d is the offset distance and M the mass of the object.

Okay so I think I'm starting to understand. In part 1 of the problem we found the moment of inertia using a mass of 6x10^24 and a radius of 6.378x10^3 and the formula I=mr^2 which gave me 9.76x10^31. For this part I would just take 9.76x10^31 + (6x10^24)(12756)^2 to get a final answer of 1.073893216E33?

You'll want to be careful about your units. What units was the radius given in? Also, check your formula for the moment of inertia of a sphere -- differently shaped bodies have different expressions (at least different constants of proportionality).

Other than those things, I don't see a problem with your method.

## 1. How is Earth's rotational inertia increase calculated at the tangent to the equator?

The increase in Earth's rotational inertia at the tangent to the equator is calculated using the formula I = mr^2, where I is the rotational inertia, m is the mass of the object, and r is the distance from the axis of rotation.

## 2. What factors affect Earth's rotational inertia increase at the tangent to the equator?

The factors that affect Earth's rotational inertia increase at the tangent to the equator include the mass and distribution of mass within the Earth, as well as the distance from the axis of rotation.

## 3. How does the rotational inertia increase at the tangent to the equator impact Earth's rotation?

The increase in rotational inertia at the tangent to the equator causes Earth to rotate at a slightly slower rate due to the conservation of angular momentum. This effect is known as the "tidal bulge" and is responsible for the Earth's tides.

## 4. Can the rotational inertia increase at the tangent to the equator change over time?

Yes, the rotational inertia increase at the tangent to the equator can change over time due to natural processes such as plate tectonics and the redistribution of mass within the Earth's interior. However, these changes are very small and have a minimal impact on Earth's rotation.

## 5. Is the rotational inertia increase at the tangent to the equator the same for all planets?

No, the rotational inertia increase at the tangent to the equator varies for different planets depending on their mass, distribution of mass, and distance from the axis of rotation. For example, Jupiter has a much larger rotational inertia increase at its equator compared to Earth due to its larger size and faster rotation.