Question

What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth can be found inside the back cover of the book.

Short answer

The rotational kinetic energy of the earth is$2.57\times {10}^{29}\text{J}$

Step-by-step solution

Step 1. Given information

The angular velocity of the earth is

$\omega =\frac{2\pi \text{rad}}{24\times 3600\text{s}}=7.27\times {10}^{-5}\text{rad}/\text{s}$

The moment of inertia of a sphere about its diameter is

$I=\frac{2}{5}{M}_{\text{earth}}{R}^2$

Step 2. Explanation

The rotational kinetic energy of the earth is

$\begin{array}{l}{K}_{\text{rot}}=\frac{1}{2}I{\omega }^2\\ K_{\text{rot}}=\frac{1}{2}\left(\frac{2}{5}{M}_{\text{earth}}{R}^2\right){\omega }^2\\ K_{\text{rot}}=\frac{1}{5}\left(5.98\times {10}^{24}\text{kg}\right){\left(6.37\times {10}^6\text{m}\right)}^2{\left(7.27\times {10}^{-5}\text{rad}/\text{s}\right)}^2\\ K_{\text{rot}}=2.57\times {10}^{29}\text{J}\end{array}$