Question

The mass of the earth is 81 times the mass of the moon and the distance between their centres is R. The distance from the centre of the earth, where gravitational force will be zero is ______.

Options

• `"R"/4`

• `"R"/2`

• `(9"R")/10`

• `"R"/81`

Answer 1

The mass of the earth is 81 times the mass of the moon and the distance between their centres is R. The distance from the centre of the earth, where gravitational force will be zero is `(9"R")/10`.

Explanation:

Let x be the distance from the centre of the earth, where the gravitational force is zero and m₀ be the point mass as shown below

Using Newton's law of gravitation,
Force between earth and point mass,

`"F"_1=("G"xx81"m"xx"m"_0)/"x"^2`   ...(i)

and force between moon and point mass,

`"F"_2=("G"xx"m"xx"m"_0)/("R"-"x")^2` ...(ii)

According to the question,

F₁ = F₂

`=> ("G"xx81"m"xx"m"_0)/"x"^2=("G"xx"m"xx"m"_0)/("R"-"x")^2`

`=> 81/"x"^2=1/("R"-"x")^2`

`=> 9/"x"=1/("R"-"x"`

`=> "x"=9/10"R"`