Using Integrals to Calculate the Rotational Energy of Earth

[matai]

So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that this is the orbital motion and I need the rotational

I found an old forum saying to use use density * Δvolume for mass, and the linear velocity with the formula KE=½mv^2. Then, integrate over the volume of the earth, i found that to be 108.321 × 10^10 km^3. So, my integral was something like this:
∫(½(465.0905584)^2(5514)dv
a=0, b=108.321 × 10^10

I ended up with 6.46005E20. I don' think is right for some reason.

I tried again with the integral:
∫(½(465.0905584)^2(5514*v)dv
a=0, b=108.321 × 10^10
and got 3.49878E32. I'm not sure if either one is right.

 

[kuruman]

I know that this is the orbital motion and I need the rotational

So you have the rotational speed of the Earth, What is an expression for the rotational energy of a spinning object? There is no need to integrate if you know the moment of inertia.

 

[Doc Al]

So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that this is the orbital motion and I need the rotational

That's the tangential speed of something at the equator due to the Earth's rotation. But different parts of the Earth will move at different speeds.

Read up on rotational velocity, inertia, and kinetic energy. Start here: Rotational Kinetic Energy

 

[Orodruin]

So you have the rotational speed of the Earth, What is an expression for the rotational energy of a spinning object? There is no need to integrate if you know the moment of inertia.

How are you planning to find the moment of inertia without integrating?

 

[matai]

So you have the rotational speed of the Earth, What is an expression for the rotational energy of a spinning object? There is no need to integrate if you know the moment of inertia.

The assignment requires an integral.

 

[matai]

How are you planning to find the moment of inertia without integrating?

So are you saying that I have found the moment of inertia, and that I need to plug in it into the KE formula?

 

[kuruman]

The assignment requires an integral.

Yes, sorry, I missed that part because I didn't read the title carefully enough.

So are you saying that I have found the moment of inertia, and that I need to plug in it into the KE formula?

Actually, I think the problem expects you to find the kinetic energy of a mass element $dm$ at radius $r$ and integrate to find the total KE.
Start from $d(KE)=\frac{1}{2}dm ~v^2$. Eventually, you will have to do a radial integral which means you will have to re-express $dm$ and $v$ in terms of the radius.