Fill in the blanks:
$(a)$ The orbital period of Mars around the Sun is $8$ times the orbital period of Mercury. If the distance of Mercury from the Sun is $5.79 \times 10^{10} \, m$,then the distance of Mars from the Sun is approximately .......
$(b)$ If the mass of an object on Earth is $m \, kg$,then the mass of the same object on the Moon is ........... .
$(c)$ The height of a geostationary satellite from the Earth's surface is approximately ........ .
$(d)$ If the distance between two objects of mass $m_1 = m_2 = 1 \, kg$ is $1 \, mm$,then the magnitude of the gravitational force between them is ........... . $[G = 6.67 \times 10^{-11} \, SI \text{ units}]$

(A) According to Kepler's third law,$T^2 \propto r^3$,so $r \propto T^{2/3}$. Given $T_M = 8 T_m$,then $r_M = r_m \times (8)^{2/3} = 5.79 \times 10^{10} \times 4 = 23.16 \times 10^{10} \, m$.
$(b)$ Mass is an intrinsic property of matter and remains constant regardless of location. Thus,the mass on the Moon is $m \, kg$.
$(c)$ $A$ geostationary satellite orbits at a height of approximately $35,800 \, km$ above the Earth's surface.
$(d)$ Using Newton's law of gravitation,$F = \frac{G m_1 m_2}{r^2}$. Here $m_1 = 1 \, kg$,$m_2 = 1 \, kg$,$r = 1 \, mm = 10^{-3} \, m$. So,$F = \frac{6.67 \times 10^{-11} \times 1 \times 1}{(10^{-3})^2} = 6.67 \times 10^{-11} \times 10^6 = 6.67 \times 10^{-5} \, N$.