The Total solution for NCERT class 6-12
(a) It is known that density ρ of air decreaseswith height y as
Where = 1.25 kg m–3 isthe density at sea level, and y0 is a constant.This density variation is called the law of atmospheres. Obtain this lawassuming that the temperature of atmosphere remains a constant (isothermalconditions). Also assume that the value of gremains constant.
(b) A large He balloon of volume 1425 m3 is usedto lift a payload of 400 kg. Assume that the balloon maintains constant radiusas it rises. How high does it rise?
[Take y0= 8000 m and = 0.18 kg m–3].
Answer - 31 : -
(a) Volume of theballoon, V = 1425 m3
Mass of the payload, m = 400 kg
Acceleration due to gravity, g = 9.8 m/s2
Density of the balloon = ρ
Height to which the balloon rises = y
Density (ρ) of air decreases with height (y) as:
This density variation is called the law of atmospherics.
It can be inferred from equation (i) that the rate ofdecrease of density with height is directly proportional to ρ,i.e.,
Where, k is the constant of proportionality
Height changes from 0 to y, while density changesfrom to ρ.
Integrating the sides between these limits, we get:
(b)
Hence, the balloon will rise to a height of 8 km.