question_answer

In the arrangements shown, a block (of mass\[m\]) is being moved up against gravity, by two identical balloons, with constant speed\[v\]. The balloons carry \[+Q\]charge each and the connecting strings are massless. \[T\] and \[B\] respectively represent tension in each of the connecting strings and buoyant force on each of the balloons. Choose the incorrect alternative.

A) \[B=\frac{mg}{2}\]                   

B) \[2T\cos \alpha =mg\]

C) \[T=\frac{{{Q}^{2}}}{16\pi {{\varepsilon }_{0}}{{l}^{2}}\sin \alpha }+\frac{mg}{2}\cos \alpha \]

D) \[2T\sin \alpha =\frac{{{Q}^{2}}}{16\pi {{\varepsilon }_{0}}{{l}^{2}}\sin \alpha }\]

Correct Answer: D

Solution :

Force between balloon,  \[F=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{Q}^{2}}}{{{\left( 2l\sin \alpha  \right)}^{2}}}\] \[=\frac{{{Q}^{2}}}{16\pi {{\varepsilon }_{0}}{{l}^{2}}{{\sin }^{2}}\alpha }\]

For the whole system, \[2B=mg,so\,B=\frac{mg}{2}\]   Also 2T \[cos=mg.\]

For horizontal direction, net force on the balloons is zero.