Question

A narrow tunnel is dug along the diameter of the earth, and a particle of mass $m_0$ is placed at $\frac{R}{2}$ distance from the centre. Find the escape speed of the particle from that place.

Answer 1

Mass density $=\frac{Me}{4/3\pi R^2}\left(P\right)$
Mass of radius $R/2=P\times \frac{\varphi }{3}\pi {\left(\frac{R}{2}\right)}^2$
$=\frac{M}{8}\left(\pi e\right)$
Gravitation equation
Escape velocity $=\sqrt{\frac{2GMe}{Req}}(Req=R/2)$
$=\sqrt{\frac{2G\left(m/8\right)}{\left(R/2\right)}}$
$V_{escape}=\sqrt{\frac{GM}{2R}}$
$\frac{1}{2}m.v{e}^2+m.[-\frac{GMe}{2R^3}\{3R^2-{\left(\frac{R}{2}\right)}^2\}]$
$\frac{1}{2}m.v{c}^2=\frac{GMe{m}_0}{2R^3}.\left\{\frac{11}{4}{R}^2\right\}$
$m.v{e}^2=m.G\frac{Me11R^2}{R^3.4}=GMem$
$m.v{e}^2=\frac{GMe}{R}.\frac{11}{4}m$
$ve=\sqrt{\frac{11}{4}\frac{GMe}{Re}}$