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

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${[\frac{{mgr}^{2}}{2 k} tanθ]}^{1 / 2}$

${[\frac{2 k}{mgr} tanθ]}^{1 / 2}$

${[\frac{mgr}{2 k} cotθ]}^{1 / 2}$

${[\frac{2 k}{{mgr}^{2}} tanθ]}^{1 / 2}$

answer is A.

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Detailed Solution

In equilibrium,
2R=mg
F_e = Tsin $θ$ ... (i)
R = Tcos $θ$ ... (ii)
From equations (i) and (iii), ... (iii)

2Tcos $θ$ = mg ... (iv)
Dividing equation (ii) by equation (iv),

$\frac{1}{2} \tan θ = \frac{F_{e}}{m g} \Rightarrow \frac{1}{2} \tan θ = \frac{k \frac{Q^{2}}{r^{2}}}{m g} \Rightarrow θ = {(\frac{m g r^{2}}{2 k} \tan θ)}^{1 / 2}$

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