Ques. 1NEET (UG) 2025
A balloon is made of a material of surface tension S and its inflation outlet (from where gas is filled in it) has small area A. It is filled with a gas of density and takes a spherical shape of radius R. When the gas is allowed to flow freely out of it, its radius r changes from R to 0 (zero) in time T. If the speed v(r) of gas coming out of the balloon depends on r as ra and T S A R then
- a = - 12, = - 12, = -1, = 12, = 72
- a = 12, = 12, = - 12, = 12, = 72
- a = 12, = 12, = -1, = +1, = 32
- a = - 12, = - 12, = -1, = - 12, = 52
Answer
(1) a = - 12, = - 12, = -1, = 12, = 72
Solution
Excess pressure inside the balloon: P = kSr (k a number)
Bernoulli at the outlet: 12 v2 = kSr v = 2kS r r-1/2, so a = - 12
Rate of loss of volume: - ddt ( 43 r3 ) = Av
-4 r2 drdt = A 2kS r-1/2
-r5/2 dr = A4 2kS dt
Integrating from r = R to 0: 27R7/2 = A4 2kS T
T S-1/2 A-1 1/2 R7/2
= - 12, = -1, = 12, = 72
Answer: (1)