Question

Two similar balloons filled with helium gas are tied to $L$ metre long strings. A body of mass $m$ is tied to another ends of the strings. The balloons float in air at distance $r$. If the amount of charge on the balloons is same, then the magnitude of charge on each balloon will be

• A. $\left[\frac{m g r^{2}}{2 k} \tan \theta\right]^{1 / 2}$
• B. $\left[\frac{2 k}{m g r^{2}} \tan \theta\right]^{1 / 2}$
• C. $\left[\frac{m g r}{2 k} \cot \theta\right]^{1 / 2}$
• D. $\left[\frac{2 k}{m g r} \tan \theta\right]^{1 / 2}$

Answer 1

Answer: (A)

In equilibrium,
$2 R =m g ........ (i) $
$F_{e} =T \sin \theta ........ (ii) $
$R =T \cos \theta$ ......... (iii)
From equations (i) and (iii),

$2 T \cos \theta=m g \ldots$(iv)
Dividing equation (ii) by equation (iv),
$ \frac{1}{2} \tan \theta=\frac{F_{e}}{m g} $
$ \Rightarrow \frac{1}{2} \tan \theta=\frac{k \frac{Q^{2}}{r^{2}}}{m g}$
$\Rightarrow \theta=\left(\frac{m g r^{2}}{2 k} \tan \theta\right)^{1 / 2}$