Question

A balloon filled with hydrogen has a volume of 1metre cube and its mass is 1kg.The volume of the block of a very light material which can just lift is (density of material of block is 91.3kg/m3, and that of air is 1.3kg/m3)

Answer 1

Dear Student,

$$\begin{gathered} \mathrm{Let}\mathrm{V}\mathrm{and}\mathrm{v}\mathrm{be}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{balloon}\&\mathrm{block}\mathrm{and}\mathrm{M}\&\mathrm{m}\mathrm{be}\mathrm{their}\mathrm{masses}. \\ \mathrm{Weight}\mathrm{of}\mathrm{air}\mathrm{displaced}=\left(\mathrm{V}+\mathrm{v}\right){\mathrm{\rho }}_{\mathrm{a}}\mathrm{g} \\ \mathrm{where},{\mathrm{\rho }}_{\mathrm{a}}=\mathrm{density}\mathrm{of}\mathrm{air} \\ \Rightarrow \left(\mathrm{V}+\mathrm{v}\right){\mathrm{\rho }}_{\mathrm{a}}\mathrm{g}=\left(\mathrm{M}+\mathrm{m}\right)\mathrm{g}...................\left(1\right) \\ \mathrm{Also},\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{v}}.............\left(2\right) \\ \mathrm{where},\mathrm{\rho }=\mathrm{density}\mathrm{of}\mathrm{block} \\ \mathrm{from}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get}, \\ \left(\mathrm{V}+\mathrm{v}\right){\mathrm{\rho }}_{\mathrm{a}}\mathrm{g}=\left(\mathrm{M}+\mathrm{\rho v}\right)\mathrm{g} \\ {\mathrm{V\rho }}_{\mathrm{a}}+{\mathrm{v\rho }}_{\mathrm{a}}=\mathrm{M}+\mathrm{\rho v} \\ \mathrm{v}=\frac{{\mathrm{V\rho }}_{\mathrm{a}}-\mathrm{M}}{\mathrm{\rho }-{\mathrm{\rho }}_{\mathrm{a}}} \\ \mathrm{v}=\frac{1\times 1.3-1}{91.3-1.3} \\ \mathrm{v}=\frac{0.3}{90}=0.0033{\mathrm{m}}^3 \end{gathered}$$