Question

A balloon filled with hydrogen has a volume of 1000 litres and its mass is 1 kg. what would be the volume of the block of very light material which it can just lift? One litre of material has mass of 91.3 gram.
(Density of air = `1.3 gram//litre)`.

Answer 1

Let V and M be the volume and mass of the balloon respectively.
Let `upsilon_(1) , m_(1)` and `rho_(1)` be the volume , mass and density of the block to be filled.
Here, `V = 1000` litres`= 1000 xx 10^(-3)m^(3) = 1m^(3)`,
`rho_(1) = 91.3 g//litre = 91.3 kg m^(-3)`,
`rho_(a) = 1.3 g//litre = 1.3 kg m^(-3)`,
Total volume of the balloon and block = `V + upsilon_(1)`.
The volume of air displaced = `(V +upsilon_(1))`. If ` rho_(a)` is teh density of air, then weight of air displaced
The balloon can lift the block if
Weight of balloon and block = weight of air displaced
i.e., `(M+m_(1))g = (V + upsilon_(1)) rho_(a)g`
`or (M+upsilon_(1)rho_(1))g = (V+upsilon_(1))rho_(a)g`
This gives, `upsilon_(1) = (Vrho_(a)-M)/(rho_(1)-rho_(a)) = (1xx1.3-1)/(91.3 = 1.3) = 0.3/90 m^(3)`
`=(0.3 xx 1000)/(90) litre = 3.33litres`.