# There is a point between the Earth and Moon where the gravitational effects of the two bodies balance each other. Calculate the distance from the center of the earth to this point.

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The force `(GM_(earth)*m)/(R_(earth))^2` of the earth on mass m, has to balance the force `(GM_(moon)*m)/(R_(moon))^2` where `R_(earth)` = distance from the center of the earth to the point and `R_(moon) ` = distance from the center of the moon to the point.

Hence, `(GM_(earth)*m)/(R_(earth))^2=(GM_(moon)*m)/(R_(moon))^2`

`rArr (M_(earth))/(R_(earth))^2=(M_(moon))/(R_(moon))^2`

`rArr R_(earth)=R_(moon)*sqrt(M_(earth)/M_(moon))` ..........(i)

Again `R_(earth)+R_(moon)` =average distance from earth to moon=384,400 km

Substituting `R_(earth)` from (i) in the above equation we get:

`R_(moon)*sqrt(M_(earth)/M_(moon))+R_(moon)=384,400`

`rArr R_(moon)(sqrt(M_(earth)/M_(moon))+1)=384,400`

Plugging in the masses of the earth and the moon we get:

`R_(moon)(sqrt((5.97*10^24)/(7.34*10^22))+1)=384,400`

`rArr R_(moon)*10.0186=384,400`

`rArr R_(moon)=38368.63 km`

Hence, `R_(earth)=346031.4 km =3.46*10^8 m`

**Therefore, the distance from the center of the earth to this point is 3.46*10^8 m.**

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