Question

Three identical small balls, each of mass $0.1\text{g}$ are suspended at one point on the silk thread having a length of $20\text{cm}$. What charges should be imparted to the balls for each thread to form an angle of $\alpha ={30}^{\circ }$ with the vertical ?

• A. $1.1\times {10}^{-8}\text{C}$
• B. $2.2\times {10}^{-8}\text{C}$
• C. $3.3\times {10}^{-8}\text{C}$
• D. $4.4\times {10}^{-8}\text{C}$

Answer 1

The correct option is C $3.3\times {10}^{-8}\text{C}$

From figure,

$a=l\cos {60}^{\circ }=20\times \frac{1}{2}=10\text{cm}$

$r=2a\cos {30}^{\circ }=2\times 10\times \frac{\sqrt{3}}{2}=10\sqrt{3}\text{cm}$

Now, applying Lami's Theorem for the equilibrium of ball, we have,

$\frac{mg}{\sin ({90}^{\circ }+{30}^{\circ })}=\frac{\sqrt{3}F}{\sin ({90}^{\circ }+{60}^{\circ })}$

$\Rightarrow F=\frac{mg}{3}$

$\Rightarrow \frac{k{q}^2}{r^2}=\frac{mg}{3}$

$\Rightarrow q=\sqrt{\frac{mg{r}^2}{3k}}$

$\Rightarrow q=\sqrt{\frac{0.1\times {10}^{-3}\times 10\times (10\sqrt{3}\times {10}^{-2}{)}^2}{3\times 9\times {10}^9}}$

$\Rightarrow q=3.3\times {10}^{-8}\text{C}$

Hence, option (c) is correct answer.

$\begin{array}{c}\text{Why this question ?}\\ \text{Concept - For any two forces (or vectors),}{F}_1\text{and}{F}_2\\ \text{net force is given by}\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos \theta },\\ \text{where}\theta \text{is angle between them.}\end{array}$