# Chapter 10 Mechanical Properties of Fluids Questions and Answers: NCERT Solutions for Class 11th Physics (Partnership Physics )

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## 31.(a) It is known that density ρ of air decreases with height y as ρ0ey/y0 . Where ρ0=1.25kgm−3 the density at sea is level, and y0 is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of the atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant.

### Ans: The given volume of the balloon is, V = 1425 m3 Mass of the payload, m=400kg And the acceleration due to gravity is, g=9.8m/s2 y0=8000m The density of helium isρHe=0.18kgm−3 , and that of air at sea level is ρ0=1.25kgm−3 Let the density of the balloon be ρ and the height to which the balloon can rise be y. The air density (ρ ) of decreases with increasing height (y ) as: ρ/ρ0=e−y/y0 This relation is also called the law of atmospherics. It can be inferred from the above law-equation that the rate of decrease of density with height is directly proportional to ρ −dρdy∝ρ ⇒−dρdy=kρ ⇒−dρρ=kdy Here, k is a proportionality constant. We set the limits such that the height changes from 0 to y , and the density varies from ρ0 to ρ . ∫ρ0ρdρρ=∫0ykdy Next, integrating the sides between these limits, we obtain: [lnρ]ρρ0=−ky ⇒ρ/ρ0=e−ky Comparing the above equation with the law of atmospherics, we get ⇒k=1y0 In conclusion, we have been able to derive the expression for the law of atmospherics. (b) A large He balloon of volume 1425m3 is used to lift a payload of400 kg . Assume that the balloon maintains a constant radius as it rises. How high does it rise? (Take y0=8000m and ρHe=0.18kgm−3 ) Ans: The generic expression for mass density is: ρ=MV In the current case, it becomes the total mass (including the payload and the helium) per unit volume of the balloon: ρ=(Mpayload+MHe)V=m+VρHeV ⇒ρ=0.46kg/m3 Using the law of atmospherics, we can obtain the value of y ρ/ρ0=e−y/y0 ⇒lnρ/ρ0=−y/y0 ⇒y=−8000×ln(0.46/1.25)=8000m=8km Therefore, the balloon can rise to a height of 8 km.

Last Updated on: November 29, 2022