A beaker contains a fluid of density ρ kg / m3, specific heat S… | JEE Main 2015 (11 Apr, Shift 1)

Q72JEE Main 2015 · 11 APR · SHIFT 1Difficult

Applications of Dimensional Analysis

A beaker contains a fluid of density $\rho$ kg / m3, specific heat S J/kgoC and viscosity $\eta$ . The beaker is filled up to height h. To estimate the rate of heat transfer per unit area ( $\dot{Q}/A$ ) by convection when beaker is put on a hot plate, a student proposes that it should depend on $\eta \cdot \left(\frac{S\Delta\theta}{h}\right)$ and $\left(\frac{1}{\rho g}\right)$ when $\Delta\theta$ (inoC) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for ( $\dot{Q}/A$ ) is:

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Answer: (a)

$[\dot{Q}/A] = \frac{ML^{2}T^{-3}}{L^{2}} = MT^{-3}$
$[\eta] = ML^{-1}T^{-1}$ , $[S\Delta\theta] = L^{2}T^{-2}$ , $[h] = L$
$\left[\eta\frac{S\Delta\theta}{h}\right] = ML^{-1}T^{-1}\cdot L^{2}T^{-2}\cdot L^{-1} = MT^{-3}$ , which already matches; multiplying by $\frac{1}{\rho g}$ would spoil it.
So the answer is (a) $\eta\frac{S\Delta\theta}{h}$ .

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