Question

The dependence of $g$ on geographical latitude at sea level is given by $g=g_0(1+\beta {\sin }^2\varphi )$ where $\varphi$ is the latitude angle and $\beta$ is a dimensionless constant. It $\Delta g$ is the error in the measurement of $g$ then the error in measurement of latitude angle is

• A. ${\Delta }_{2\varphi }=\frac{3\Delta g}{g_0\beta \sin \left(2\varphi \right)}$
• B. ${\Delta }_{\varphi }=\frac{\Delta g}{g_0\beta \cos \left(2\varphi \right)}$
• C. ${\Delta }_{\varphi }=\frac{\Delta g}{g_0}$
• D. Zero

Answer 1

The correct option is B ${\Delta }_{\varphi }=\frac{\Delta g}{g_0\beta \cos \left(2\varphi \right)}$

$g=g_o(1+\beta {\sin }^2\varphi )$

$dg=g_o\beta \cdot 2\sin \varphi \cdot \cos \varphi d\varphi$ (Differentiate)

$d\varphi =\frac{dg}{g_o\beta \sin 2\varphi }$

$\Delta \varphi =\frac{\Delta g}{g_o\beta \sin 2\varphi }$.