Question

Two similar balloons filled with helium gas are tied to $Lm$ long strings. A body of mass m is tied to another ends of the strings. The balloons float on 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. ${\left[\frac{mgr}{2k}\cot \theta \right]}^{1/2}$
• B. ${\left[\frac{2k}{mg{r}^2}\tan \theta \right]}^{1/2}$
• C. ${\left[\frac{mg{r}^2}{2k}\tan \theta \right]}^{1/2}$
  
  
• D. ${\left[\frac{2k}{mgr}\tan \theta \right]}^{1/2}$

Answer 1

The correct option is C ${\left[\frac{mg{r}^2}{2k}\tan \theta \right]}^{1/2}$



If $R$ is the buoyant force acting on the two balloons due to air then,
Diagram

In equilibrium
$2R=mg\dots \left(i\right)$
$F_e=T\sin \theta \dots \left(ii\right)$
$R=T\cos \theta \dots \left(iii\right)$
From equations $\left(i\right)$ and $\left(iii\right)$
$2T\cos \theta =mg\dots \left(iv\right)$
Dividing equation $\left(ii\right)$ by equation $\left(iv\right)$,
$\frac{1}{2}\tan \theta =\frac{F_e}{mg}$
$\Rightarrow \frac{1}{2}\tan \theta =(\frac{k\frac{Q^2}{r^2}}{mg}$)
$\Rightarrow Q={\left(\frac{mg{r}^2}{2k}\tan \theta \right)}^{1/2}$

Hence. Option (a) is correct.