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Question

(A) Let the expression for velocity be $V = k T^a A^b \rho^c$,where $k$ is a dimensionless constant. Writing the dimensions of each quantity: $[V] = [

Vedclass · Accepted Answer

$V = k \sqrt{\frac{T}{A \rho}}$

Vedclass · Answer

(A) Let the expression for velocity be $V = k T^a A^b \rho^c$,where $k$ is a dimensionless constant. Writing the dimensions of each quantity: $[V] = [LT^{-1}]$,$[T] = [MLT^{-2}]$,$[A] = [L^2]$,$[\rho] = [ML^{-3}]$. Substituting these into the equation: $[LT^{-1}] = [MLT^{-2}]^a [L^2]^b [ML^{-3}]^c = [M^{a+c} L^{a+2b-3c} T^{-2a}]$. Comparing the powers of $M$,$L$,and $T$ on both sides: For $M$: $a + c = 0 \Rightarrow c = -a$. For $T$: $-2a = -1 \Rightarrow a = 1/2$. Thus,$c = -1/2$. For $L$: $a + 2b - 3c = 1$. Substituting $a = 1/2$ and $c = -1/2$: $1/2 + 2b - 3(-1/2) = 1 \Rightarrow 1/2 + 2b + 3/2 = 1 \Rightarrow 2 + 2b = 1 \Rightarrow 2b = -1 \Rightarrow b = -1/2$. Substituting the values of $a, b, c$ back into the expression: $V = k T^{1/2} A^{-1/2} \rho^{-1/2} = k \sqrt{\frac{T}{A \rho}}$.

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