Question

The center of a moon of mass m is a distance D from the center of a planet of mass M. At some distance x from the center of the planet, along a line connecting the centers of planet and moon, the net force on an object will be zero. a) Derive an expression for x in terms of m, M, and D. b) If the net force is zero a distance ⅔D from the planet, what is the ratio R of the mass of the planet to the mass of the moon, M/m?

Answer 1

Answer with Explanation:

Let  rest mass $m_0$ at point P  at  distance x from center of the planet, along a line connecting the centers of planet and the moon.

Mass of moon=m

Distance between the center of moon and center of planet=D

Mass of planet=M

We are given that net force on an object will be zero

a.We have to derive an expression for x in terms of m, M and D.

We know that gravitational force=$\frac{GmM}{r^2}$

Distance of P from moon=D-x

$F_m$=Force applied on rest mass due to m

$F_m$=Force on rest mass due to mas M

$F_M=F_m$ because net force is equal to 0.

$F_m=F_M$

$\frac{Gm_0m}{(D-x)^2}=\frac{Gm_0M}{x^2}$

$\frac{m}{(D-x)^2}=\frac{M}{x^2}$

$\frac{x^2}{(D-x)^2}=\frac{M}{m}$

$\frac{x}{D-x}=\sqrt{\frac{M}{m}}$

Let $R=\sqrt{\frac{M}{m}}$

Then, $\frac{x}{D-x}=R$

$x=DR-xR$

$x+xR=DR$

$x(1+R)=DR$

$x=\frac{DR}{1+R}$

b.We have to find the ratio R of the mass of the mass of the planet to the mass of the moon when x=$\frac{2}{3}D$

Net force is zero

$F_m=F_M$

$\frac{Gm_0m}{(D-\frac{2}{3}D)^2}=\frac{Gm_0M}{\frac{4}{9}D^2}$

$\frac{m}{\frac{D^2}{9}}=\frac{9M}{4D^2}$

$\frac{M}{m}=4$

Hence, the ratio R of the mass of the planet to the mass of the moon=4:1