The Rotational Kinetic Energy of the Earth

In summary: This is why the moment of inertia of the Earth is smaller than expected, as the densest materials have sunk to the center, reducing the overall moment of inertia.
  • #1
soupastupid
34
0

Homework Statement



The Earth can be approximated as a sphere of uniform density, rotating on its axis once a day. The mass of the Earth is 5.97×10^24 kg, the radius of the Earth is 6.38×10^6 m, and the period of rotation for the Earth is 24.0 hrs.

What is the moment of inertia of the Earth? Use the uniform-sphere approximation described in the introduction.

I = (2/5)(5.97×10^24 kg)(6.38×10^6 m)^2
I = 9.720×10^37 kg m^2

why is it multiplied by (2/5) ?? is that value just given by a uniform sphere equation?

Consider the following statements, all of which are actually true, and select the one that best explains why the moment of inertia of the Earth is actually smaller than the moment of inertia you calculated.

A) The Earth is an oblate spheroid rather than a perfect sphere. For an oblate spheroid, the distance from the center to the equator is a little larger than the distance from the center to the poles. This is a similar shape to a beach ball resting on the ground, being pushed on from above.

B) The Earth does not have uniform density. As the planet formed, the densest materials sank to the center of the Earth. This created a dense iron core. Meanwhile, the lighter elements floated to the surface. The crust of the Earth is considerably less dense than the core.

C) While the Earth currently has a period of 24 hours, it is in fact slowing down. Once it was rotating much faster, giving days that were closer to 20 hours than 24 hours. In the future, it is expected that days will become longer.

Homework Equations



?

The Attempt at a Solution



The moment is lower because the mass is concentrated in the center in of the earth. It is not equally dense?

so its B right?
 

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  • #2
soupastupid said:
why is it multiplied by (2/5) ?? is that value just given by a uniform sphere equation?
Yes. An object's moment of inertia depends on how its mass is distributed--different shapes will have different moments of inertia. See: http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#cmi"

The Attempt at a Solution



The moment is lower because the mass is concentrated in the center in of the earth. It is not equally dense?

so its B right?
Good. The closer an object's mass is to the axis of rotation, the smaller its moment of inertia.
 
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  • #3


Yes, B is the best explanation for why the moment of inertia of the Earth is actually smaller than the calculated value. The moment of inertia is a measure of an object's resistance to rotational motion, and it is affected by the distribution of mass within the object. In the case of the Earth, the dense iron core and the less dense crust create a non-uniform distribution of mass, resulting in a smaller moment of inertia than if the Earth were a perfect uniform sphere. This is also why the (2/5) factor is used in the calculation of the moment of inertia for a uniform sphere, as it takes into account the non-uniform distribution of mass.
 

1. What is the rotational kinetic energy of the Earth?

The rotational kinetic energy of the Earth is the energy associated with its rotation around its own axis. It is a measure of the Earth's rotational motion and is dependent on its mass, shape, and rotational speed.

2. How is the rotational kinetic energy of the Earth calculated?

The rotational kinetic energy of the Earth can be calculated using the formula KE = (1/2)Iω^2, where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity. The moment of inertia takes into account the Earth's mass and distribution of mass around its axis of rotation.

3. What factors affect the rotational kinetic energy of the Earth?

The rotational kinetic energy of the Earth is affected by its mass, distribution of mass, and rotational speed. The larger the Earth's mass and the faster it rotates, the higher its rotational kinetic energy will be. Additionally, any changes in its shape or distribution of mass can also impact its rotational kinetic energy.

4. How does the rotational kinetic energy of the Earth impact our daily lives?

The rotational kinetic energy of the Earth is responsible for the Earth's rotation, which in turn creates the day and night cycle. It also impacts the Earth's magnetic field, ocean currents, and weather patterns. Additionally, it plays a role in the Earth's stability and helps to maintain its tilt, which is responsible for the changing seasons.

5. Can the rotational kinetic energy of the Earth change over time?

Yes, the rotational kinetic energy of the Earth can change over time. This can be due to factors such as the redistribution of mass through plate tectonics, changes in the Earth's rotational speed, or even external forces such as meteorite impacts. However, these changes are typically very small and occur over long periods of time.

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