Question

Two identical spheres, touching each other, are placed on a rough horizontal surface of static friction co-efficient ${\mu }_s$ as shown in figure. An another identical sphere of same mass and same radius is also placed on the spheres symmetrically. The contact surface between sphere C, with other spheres is smooth. What should be the minimum value of ${\mu }_s$, so that the system can stay in equilibrium?

• A. $\frac{1}{2\sqrt{3}}$
• B. $\frac{1}{3\sqrt{3}}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{1}{2\sqrt{2}}$

Answer 1

The correct option is B $\frac{1}{3\sqrt{3}}$

Referring the figure, $2Ncos{30}^o=mg\Rightarrow N=\frac{mg}{2cos{30}^o}$
Balancing the force on sphere A, $\mu (mg+Ncos{30}^o)=Nsin{30}^o$
Solving we get, $\mu =\frac{1}{3\sqrt{3}}$