Question

Three uniform sphere of mass M and radius R each are kept in such a way that each touches the other two. The magnitude of the gravitational force on any of the sphere due to the other two is

• A. $\frac{\sqrt{3}G{M}^2}{4R^2}$
• B. $\frac{3G{M}^2}{2R^2}$
• C. $\frac{\sqrt{3}G{M}^1}{R^2}$
• D. $\frac{\sqrt{5}G{M}^2}{4R^2}$

Answer 1

The correct option is A $\frac{\sqrt{3}G{M}^2}{4R^2}$

Force between two spheres will be

$F_1=F_2=\frac{GMM}{{\left(2R\right)}^2}=\frac{G{M}^2}{4R^2}$

Two forces of equal magnitude are acting at angle ${60}^o$ on any of the sphere

$F_{net}=\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos {60}^o}=F\sqrt{3}$

$F_{net}=\frac{\sqrt{3}}{4}\frac{G{M}^2}{R^2}$

.Hence, magnitude of force is $F_{net}=\frac{\sqrt{3}}{4}\frac{G{M}^2}{R^2}$