The Earth's rotational kinetic energy is the kinetic Energy that the Earth
has due to rotation.
The rotational kinetic energy of the Earth is approximately <u>3.331 × 10³⁶ J</u>
Reasons:
<em>The parameters required for the question are; </em>
<em>Mass of the Earth, M = </em><em>5.97 × 10²⁴ kg</em>
<em>Radius of the Earth, R = </em><em>6.38 × 10⁶ m</em>
<em>The rotational period of the Earth, T = </em><em>24.0 hrs</em><em>.</em>
![The \ moment \ of \ inertia \ of \ uniform \ sphere \ is \ I = \mathbf{\dfrac{2}{5} \cdot M \cdot R^2}](https://tex.z-dn.net/?f=The%20%5C%20moment%20%20%5C%20of%20%5C%20%20inertia%20%5C%20%20of%20%5C%20%20uniform%20%5C%20%20sphere%20%5C%20%20is%20%5C%20I%20%3D%20%20%20%5Cmathbf%7B%5Cdfrac%7B2%7D%7B5%7D%20%5Ccdot%20M%20%5Ccdot%20R%5E2%7D)
Which gives;
![\mathbf{I_{Earth}} = \dfrac{2}{5} \times 5.97 \times 10 ^{24} \cdot \left(6.38 \times 10^6 \right)^2 = 9.7202107 \times 10^{37}](https://tex.z-dn.net/?f=%5Cmathbf%7BI_%7BEarth%7D%7D%20%3D%20%20%20%5Cdfrac%7B2%7D%7B5%7D%20%5Ctimes%205.97%20%5Ctimes%2010%20%5E%7B24%7D%20%5Ccdot%20%5Cleft%286.38%20%5Ctimes%2010%5E6%20%5Cright%29%5E2%20%3D%209.7202107%20%5Ctimes%2010%5E%7B37%7D)
![\mathrm{The \ rotational \ kinetic \ energy \ is} \ E_{rotational} = \mathbf{\dfrac{1}{2} \cdot I \cdot \omega^2}](https://tex.z-dn.net/?f=%5Cmathrm%7BThe%20%5C%20rotational%20%5C%20%20kinetic%20%20%5C%20energy%20%5C%20%20is%7D%20%5C%20%20%20E_%7Brotational%7D%20%3D%20%5Cmathbf%7B%5Cdfrac%7B1%7D%7B2%7D%20%5Ccdot%20I%20%5Ccdot%20%5Comega%5E2%7D)
![\mathrm{The \ angular \ speed, \ \omega} = \mathbf{\dfrac{2 \dcdot \pi}{T}}](https://tex.z-dn.net/?f=%5Cmathrm%7BThe%20%5C%20angular%20%5C%20speed%2C%20%5C%20%5Comega%7D%20%3D%20%5Cmathbf%7B%5Cdfrac%7B2%20%5Cdcdot%20%5Cpi%7D%7BT%7D%7D)
Therefore;
![\omega = \dfrac{2 \cdot \pi}{24} = \dfrac{\pi}{24}](https://tex.z-dn.net/?f=%5Comega%20%3D%20%5Cdfrac%7B2%20%5Ccdot%20%5Cpi%7D%7B24%7D%20%20%3D%20%5Cdfrac%7B%5Cpi%7D%7B24%7D)
Which gives;
![\mathbf{E_{rotational}} = \dfrac{1}{2} \times 9.7202107 \times 10^{37} \times \left( \dfrac{\pi}{12} \right)^2 = 3.331 \times 10^{36}](https://tex.z-dn.net/?f=%5Cmathbf%7BE_%7Brotational%7D%7D%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20%209.7202107%20%5Ctimes%2010%5E%7B37%7D%20%5Ctimes%20%20%5Cleft%28%20%20%5Cdfrac%7B%5Cpi%7D%7B12%7D%20%5Cright%29%5E2%20%3D%203.331%20%5Ctimes%2010%5E%7B36%7D)
The rotational kinetic energy of the Earth,
= <u>3.331 × 10³⁶ Joules</u>
Learn more here:
brainly.com/question/13623190
<em>The moment of inertia from part A of the question (obtained online) is that of the Earth approximated to a perfect sphere</em>.
<em>Mass of the Earth, M = 5.97 × 10²⁴ kg</em>
<em>Radius of the Earth, R = 6.38 × 10⁶ m</em>
<em>The rotational period of the Earth, T = 24.0 hrs</em>