The volume $V$ of water passing any point of a uniform tube during $t$ seconds is related to the…
The volume $V$ of water passing any 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 $V \propto A^{\alpha} u^{\beta} t^{\gamma}$ ($\alpha,, \beta, \gamma \neq 1$)
which one of the following will be true? ($\alpha, \beta ~\&~ \gamma$ are integers)
$\alpha=\beta=\gamma$
$\alpha \neq \beta=\gamma$
$\alpha=\beta \neq \gamma$
$\alpha \neq \beta \neq \gamma$
Solution
The dimension of the two sides of proportionality are
$L^{3}=L^{2 \alpha}\left(L T^{-1}\right)^{\beta} T^{\gamma}=L^{2 n+\beta} T^{\gamma-\beta}$
Equating the powers of dimensions on both sides, we have
$2 \alpha+\beta=3$
$\gamma-\beta=0$
which give $\beta=\gamma$ and $\alpha=\frac{1}{2}(3-\beta)$
i.e., $\alpha \neq \beta=\gamma$.
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Asked in: JEE Mains - Units and Dimensions - Test 3