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.…

Question

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 ρ 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 in time T. If the speed v(r) of gas coming out of the balloon depends on r as r^a and T S^αA^βρ^γR^δ, then:

1. A. a=12,\ α=12,\ β=-1,\ γ=+1,\ δ=32
2. B. a=-12,\ α=-12,\ β=-1,\ γ=-12,\ δ=52
3. C. a=-12,\ α=-12,\ β=-1,\ γ=12,\ δ=72 (Correct answer)
4. D. a=12,\ α=12,\ β=-12,\ γ=12,\ δ=72

Correct Answer

Option C — a=-12,\ α=-12,\ β=-1,\ γ=12,\ δ=72

Detailed Solution & Explanation

The correct answer is Option C.

Key Points

• Dimensional analysis of $T\propto S^{\alpha}A^{\beta}\rho^{\gamma}R^{\delta}$: writing $[S]=MT^{-2}$, $[A]=L^2$, $[\rho]=ML^{-3}$, $[R]=L$ and matching $M^0L^0T^1$.
• Time exponent: $-2\alpha=1\Rightarrow\alpha=-\tfrac{1}{2}$; mass exponent: $\alpha+\gamma=0\Rightarrow\gamma=\tfrac{1}{2}$.
• Length exponent: $2\beta-3\gamma+\delta=0$; with $\beta=-1$ (from $A$) this gives $\delta=\tfrac{7}{2}$.
• The outflow speed set by surface-tension pressure gives $v\propto r^{-1/2}$, so $a=-\tfrac{1}{2}$.