Question

Radius of moon is 1//4 times that of earth and mass is 1/81 times that of earth. The point at which gravitational field due to earth becomes equal and opposite to that of moon, is (Distance between centres of earth and moon is 60R, where R is radius of earth

Answer 1

Given :

• Radius of earth = R
• Distance between centers of Earth and moon = 60 R
• Radius of moon = R/4
• Mass of Earth = M
• Mass of moon = M/81

To find :

• Point between Earth and moon at which gravitational force due to earth will be equal and opposite to that of the moon

Solution :

• The point is somewhere between the Earth and the moon. Let the distance of the point from the center of the Earth be 'x'.
• Hence, the distance of that point from the center of the moon will be (60 - x) .
• We know that,

        $Gravitational field = \frac{GM}{R^{2} }$

• At that point, the gravitational field due to the earth and the moon will be equal in magnitude but opposite in direction.
• ∴   $\frac{GM}{x^{2} } = \frac{G\frac{M}{81} }{(60R-x)^{2} }$    
• ∴   $x^{2} = 81(60R - x)^{2}$
• Taking square root,

           $x = 9(60R-x)$

       ∴  $x = 540R - 9x$

       ∴  $10x = 540R$

       ∴   x = 54R

Answer : The point at which gravitational field due to earth becomes equal and opposite to that of the moon will be at a distance of 54R from center of earth.