Question

Two point masses $M$ are kept fixed on the $x$ -axis at a distance $a$ from the origin, another point mass $m$ is moving in a circular path of radius $R$ (in $y-z$ plane) under the influence of gravitational force of attraction, then speed of $m$ will be

(Assume no forces are acting on $m$ other than the gravitational forces by two $M$ )

• A. $\sqrt{\frac{2 G M R^{2}}{\left(R^{2} + a^{2}\right)^{\frac{3}{2}}}}$
• B. $\sqrt{\frac{G M R^{2}}{\left(R^{2} + a^{2}\right)^{\frac{3}{2}}}}$
• C. $\sqrt{\frac{3 G M R^{2}}{\left(R^{2} + a^{2}\right)^{\frac{3}{2}}}}$
• D. $\sqrt{\frac{5 G M R^{2}}{\left(R^{2} + a^{2}\right)^{\frac{3}{2}}}}$

Answer 1

Answer: (A)

$F_{R}=2Fsin \theta $
$\frac{m v^2}{R}=(2)\left(\frac{G M m}{r^2}\right)\left(\frac{R}{r}\right)$
$r=\sqrt{a^{2} + R^{2}}$
$\Rightarrow v=\sqrt{\frac{2 G M R^{2}}{\left(R^{2} + a^{2}\right)^{\frac{3}{2}}}}$