1.At what depth $d$ below Earth's surface does acceleration due to gravity become equal to its value at height $h=R/2$?
Sol: $g_h = g \left(\frac{R}{R + R/2}\right)^2 = \frac{4}{9}g$. $g_d = g\left(1 - \frac{d}{R}\right) = \frac{4}{9}g \implies 1 - \frac{d}{R} = \frac{4}{9} \implies d = \frac{5}{9}R$.
2.A satellite of mass $m$ revolves in a circular orbit of radius $r$ around Earth of mass $M$. Find its binding energy and energy required to shift it to orbit of radius $2r$.
Sol: Total Energy $E_1 = -\frac{GMm}{2r} \implies$ Binding Energy = $\frac{GMm}{2r}$. Energy required $\Delta E = E_2 - E_1 = -\frac{GMm}{4r} - \left(-\frac{GMm}{2r}\right) = \frac{GMm}{4r}$.
3.Calculate the gravitational potential and gravitational field at a distance $r$ inside and outside a uniform solid sphere of mass $M$ and radius $R$.
Sol: Outside ($r \ge R$): $V = -\frac{GM}{r}, E = \frac{GM}{r^2}$. Inside ($r < R$): $E = \frac{GM r}{R^3}, V = -\frac{GM}{2R^3}(3R^2 - r^2)$.
