(a) Calculate the kinetic energy that the earth has because of its rotation about its own axis. Assume that the earth is a uniform sphere and that its path around the sun is circular. For comparison, the total energy used in the United States in one year is about 9.33e9 Joules.

I found moment of inertia by .5xMRe2

How do you find angular velocity?

The moment of inertia of a uniform sphere is NOT (1/2) M R^2. Try that part again.

http://scienceworld.wolfram.com/physics/MomentofInertiaSphere.html

What you want to calculate is
Rotational KE = (1/2) I w^2

w is the angular velocity for ROTATION in radians per second, and equals 2 pi divided by the number of seconds in a sidereal day (23 hours, 56 minutes)

The circular path assumption does NOT have to be made. They are either trying to confuse you or the teacher who assigned the question is not well trained.

To find the angular velocity (ω) of the Earth, we can use the formula:

ω = 2π / T

Where:
- ω represents the angular velocity
- π is the mathematical constant pi (approximately 3.14159)
- T is the period of rotation of the Earth, which is approximately 24 hours or 86,400 seconds.

Substituting the values into the formula, we can calculate the angular velocity of the Earth:

ω = 2π / 86,400 seconds

Calculating this equation gives us the angular velocity of the Earth.

Once you have the angular velocity, you can proceed with calculating the kinetic energy of the Earth's rotation about its own axis using the formula:

Kinetic Energy = (1/2) * I * ω^2

Where:
- Kinetic Energy represents the energy of rotation
- I is the moment of inertia of the Earth, which you have already calculated as 0.5 * M * R^2 (where M represents the mass of the Earth and R represents the radius of the Earth)
- ω is the angular velocity of the Earth

Substituting the known values into the formula, you can calculate the kinetic energy of the Earth's rotation about its own axis.

Keep in mind that the energy calculated above is only for the Earth's rotation about its own axis, and not for its path around the sun.