Question

A small body of super dense material, whose mass is half of the earth (but whose size is very small compared to the size of the earth), starts from rest at a height $H$ above the earths surface, and reaches the earths surface in time $t$. Calculate time $t$ assuming that $H$ is very small compared to the radius of the earth. Acceleration due to gravity near the surface of the earth is $g$.

Answer 1

Let $m_e$ be mass of earth

centre of mass w.r.t earth = $\frac{m\times d+m_e\left(0\right)}{m+m_e}$

Since, $m$ = $\frac{m_e}{2}$

$\therefore c.o.m$ = $\frac{\frac{m_e}{2}\times H}{\frac{3m_e}{2}}$

= $\frac{H}{3}$

actual distance covered = $H$ - $\frac{H}{3}$

= $\frac{2H}{3}$

S =$ut+\frac{1}{2}a{t}^2$

$\frac{2H}{3}=0+\frac{1}{2}\times g\times t^2$

t = $\sqrt{\frac{4H}{3g}}$