Question

lmagine a vertical mine shaft of depth x metres located near the equator (of earth). A plumb line is hanging in the shaft and a stone is dropped from rest next to it. The stone will not fall parallel to the plumb line, but will deviate in easterly direction by an amount of y metres.
Given $\omega$ is angular speed of earth and $g$ is acceleration due to gravity)
If y$=\frac{\omega g}{n}{\left(\frac{2x}{g}\right)}^{\frac{3}{2}}$ find n.

Answer 1

Let, 'v' be the velocity attained by stone at time 't'. Thus, acceleration of the body in east direction due to coriolis effect is $2\omega v$. Integrating this acceleration twice w.r.t time, we get the displacement of stone 'y'. Hence,

$y=2\omega vv=gtx=\frac{g{t}^2}{2}\Rightarrow y=2\omega gt\iint ydt=\frac{\omega g{t}^3}{3}\Rightarrow y=\frac{\omega g}{3}{\left(\frac{2x}{g}\right)}^{\frac{3}{2}}\Rightarrow n=3$