Answer to Question #173955 in Astronomy | Astrophysics for Mankaran

Question

(i)           The resultant gravitational force exerted upon the body at this point is zero. Calculate the distance R of the body from the centre of the Earth given that

r = 3.9 × 107 m and M_E = 81 M_M

Answer 1

The two forces are in equilibrium, which means the force of attraction between the earth and body (which is r = 3.9 × 10⁷ away from Earth) is the same as the force between the body and mass M_M at distance R:

$$F_E=F_M,\\\space\\ G\frac{M_Em}{r^2}=G\frac{M_Mm}{R^2},\\\space\\ R=r\sqrt{\frac{M_M}{M_E}}=3.9·10^7\sqrt{\frac{1}{81}}=4.3·10^6\text{ m}.$$