Calculate the distance d from the center of the earth at whi

Calculate the distance d from the center of the earth at which the force on a particle from the moon is equal to 0.66 times the force on the particle from the earth. The particle is restricted to the line through the center of the earth and the moon. Justify the two solutions (d_1,

Solution

Let the mass of the particle be m

Me = mass of earth = 5.97*10^24 kg

Mm = mass of moon = 7.35*10^22 kg

D = distance between earth and moon = 3.844*10^8 m

So, according to the equation:

G*Me*m/r^2 = G*Mb*m/(D-r)^2 <---------- r = distance of the particle from earth

So, Me/r^2 = Mb/(D-r)^2

So, 5.97*10^24/r^2 = 7.35*10^22/(3.844*10^8 - r)^2

So, solving the equation we get ,

two solutions :

r1 = 3.46*10^8 m = 1.82*10^5 km and

r2 = 4.32*10^9 m = 4.32*10^5 km

So r1 < r2

These are the two solutions.

First one lies between the line joining the centers of the two bodies and thus to the left of moon

The second point lies to the right of moon.