Chapter 1.6, Problem 55ES

In the study of falling objects near the surface of the Earth, the acceleration $g$ due to gravity is commonly taken to be a constant $9.8{\text{ m/s}}^2$ . However, the elliptical shape of the Earth and other factors cause variations in this value that depend on latitude. The following formula, known as the World Geodetic System $1984$ (WGS $84$ ) Ellipsoidal Gravity Formula, is used to predict the value of $g$ at a latitude of $\varphi$ degrees (either north or south of the equator):

$g=9.7803253359\frac{1+0.0019318526461{\sin }^2\varphi }{\sqrt{1-0.0066943799901{\sin }^2\varphi }}{\text{m/s}}^2$

(a) Use a graphing utility to graph the curve $y=g\left(\varphi \right)$ for $0°\le \varphi \le 90°$ . What do the values of $g$ at $\varphi =0°$ and at $\varphi =90°$ tell you about the WGS $84$ ellipsoid model for the Earth?

(b) Show that $g=9.8{\text{ m/s}}^2$ somewhere between latitudes of $38°$ and $39°$ .