Question

The volume V of water passing through a point of a uniform tube during t seconds is related to the cross-sectional area A of the tube and velocity u of water by the relation $V \propto A^{\alpha}u^{\beta}t^{\gamma}$, which one of the following will be true

• A. $\alpha = \beta = \gamma$
• B. $\alpha \neq \beta = \gamma$
• C. $\alpha = \beta \neq \gamma$
• D. $\alpha \neq \beta \neq \gamma$

Answer 1

Answer: (B)

$\alpha \neq \beta = \gamma$

Answer 2

Writing dimensions of both sides

$\lbrack L^{3}\rbrack = \lbrack L^{2}\rbrack^{\alpha}\lbrack LT^{- 1}\rbrack^{\beta}\lbrack T\rbrack^{\gamma} \Rightarrow \lbrack L^{3}T^{0}\rbrack = \lbrack L^{2\alpha + \beta}T^{\gamma - \beta}\rbrack$

By comparing powers of both sides $2\alpha + \beta = 3$ and $\gamma - \beta = 0$

Which give $\beta = \gamma$ and $\alpha = \frac{1}{2}(3 - \beta)$ i.e. $\alpha \neq \beta = \gamma$.