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

• A. $\frac{GMm{(1+\sqrt{3})}^2}{L^2}$
• B. $\frac{GMm}{L^2{(1+\sqrt{3})}^{}}$
• C. $\frac{GMm{(1-\sqrt{3})}^2}{L^2}$
• D. $\frac{GMm{(1+\sqrt{3})}^{}}{L^2}$

Answer 1

Answer: (A)

Net force on any mass at the position between two masses is zero

Let say the distance between M and m is "x" so the net force due to M and 3M on mass m is zero

so we have,

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

now by solving above equation

$L-x=\sqrt{3}x$

$\to x=\frac{L}{1+\sqrt{3}}$

now by the formula of force between two mass

$F=\frac{GMm}{x^2}$

plug in the value of "x" in it

$F=\frac{GMm}{{\left(\frac{L}{1+\sqrt{3}}\right)}^2}$

$F=\frac{GMm(\sqrt{3}+1{)}^2}{L^2}$