Q$3(20$pt$)$: centrifugal force

mass and radius of earth are $M_E=5\mathrm{.}97{10}^{24}Kg$ and $R_E=6\mathrm{.}370{10}^{6}m,$ angular speed of earth spin motion $=7\mathrm{.}2722{10}^{-5}Ra\frac{d}{s}.$ latitude of New York is $=40\mathrm{.}71$deg, colatitude $=90-.$

Apparent weight vec$(F{)}_{\text{e}\text{f}\text{f}}$ of mass $m$ at colatitude $$ is

vec$(F{)}_{\text{e}\text{f}\text{f}}=-$widehat$(r)m{g}_0+$widehat$()m{}^2{R}_{E}sin$

$,g_0=\frac{G{M}_E}{R_E^2}$ is gravitation acceleration at surface of earth, widehat$(r)$ is unit vector from a point on earth surface toward center of earth. make a plane perpendicular to the axis from south pole to north pole at colatitude $,$widehat$()$ is the outward unit vector from cross point.

$($a $10$pt$)$ For a mass $m=50Kg$ at New York, find apparent weight $F_{\text{e}\text{f}\text{f}}$ to $1$ st order of $\frac{{}^2{R}_E}{g_0},$

$($b $10$pt$)$ find angle $$ between vec$(F{)}_{\text{e}\text{f}\text{f}}$ and $-$widehat$(r)[$direction of true gravitation force$]$ to $1$ st order of $\frac{{}^2{R}_E}{g_0}.$