$($a$)$Write an expression for the force exerted by the Moon, mass $\backslash ($ M $\backslash ),$ on a particle of water, mass $\backslash ($ m $\backslash )$ on the earth $\backslash ($ s $\backslash )$ surface directly beneath the moon and lying on the line connecting the center of the earth with the center of the Moon. Take the radius of the earth to be $\backslash ($ R $\backslash ),$ and the distance between the center of the earth and the center of the Moon to be $\backslash ($ r $\backslash ).($b$)$ Suppose that the particle of water was instead at the center of the earth. What force would the Moon exert on it there? $($c$)$ Show that the difference in these forces is given by

$\backslash [$

F$\_\{$T$\}=\backslash$frac$\{2$ G M m R$\}\{$r$^\{3\}\}$

$\backslash ]$

And represents the tidal force, the force on water, relative to the earth, caused by the gravitational pull of the moon. What is the direction of the tidal force? $($d$)$ Repeat for a particle on the far side of the earth, lying on the same line connecting the earth$-$Moon centers. What the direction of this tidal force? $($e$)$ Explain why there are two tidal bulges in the oceans $($and in the earth s crust$),$ one pointing toward the Moon, and the other away from it$.$