Answer to Question #289764 in Physics for Nddd

Question

Determine the location where an object with mass m0 when placed between the moon and earth experiences a gravitational force from the moon equal to a third of the gravitational force from earth. The distance from the earth to moon is 384,400 km. The ratio of mases between moon and earth is 0.0123. The radii of moon and earth are 1738.1 km and 6378.1 km, respectively.

Answer 1

The location can be found as follows:

$$G\dfrac{mm_0}{(D-x)^2}=G\dfrac{Mm_0}{x^2},\\\space\\ \dfrac{m}{(D-x)^2}=\dfrac{M}{x^2},\\\space\\ x^2m=MD^2-2MDx+Mx^2,\\ x^2(M-m)-x·2MD+MD^2=0,\\\space\\ x=322900\text{ km}$$

from the Earth. This point is also known as Lagrange point.