Question

two point masses M and 3M are placed at a distance L apart. Another point mass m is placed in between on the line joining them so that the net gravitational force acting on it due to masses M and 3M is zero. The magnitude of gravitational force acting due to mass M on mass m will be

Answer 1

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here is your answer

Answer 2

Answer: $F=G\frac{Mm(1+\sqrt3)^2}{L^2}$

Explanation:

Let the mass m be located at a distance x from mass M. Therefore, the distance between m and 3M is L-x.

The magnitude of gravitational force between two masses is

$F=G\frac{M_1M_2}{R^2}$

The gravitational force between mass M and m is:

$F_1=\frac{Mm}{x^2}$

The gravitational force between 3M and m is

$F_2=\frac{3Mm}{(L-x)^2}$

The net force on mass m is zero due to mass M and 3M

$\Rightarrow F_1=F_2\\ \Rightarrow G\frac{M_1M_2}{R^2}=\frac{3Mm}{(L-x)^2}\\ \Rightarrow (L-x)^2=3x^2\\ \Rightarrow x=\frac{L}{(1+\sqrt3)}$ Ignoring the negative value.

The magnitude of gravitational force acting due to mass M on m will be:

$F_1=G\frac{Mm}{x^2}=\frac{Mm(1+\sqrt3)^2}{L^2}$