JEE Main 2026 Jan 23 Shift 2, Physics Q23: Hydrodynamic

JEE Main2026Jan 23, Shift 2Physics

A ball of radius $r$ and density $ρ$ dropped through a viscous liquid of density $σ$ and viscosity $η$ attains its terminal velocity at time $t$ , given by $t = A ρ^{a} r^{b} η^{c} σ^{d}$ , where $A$ is a constant and $a, b, c$ and $d$ are integers. The value of $\frac{b + c}{a + d}$ is ______.

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Solution

Dimensional analysis using $[ρ] = M L^{- 3}$ , $[r] = L$ , $[η] = M L^{- 1} T^{- 1}$ , $[σ] = M L^{- 3}$ , $[t] = T$ .

$T = M^{a + c + d} \cdot L^{- 3 a + b - c - 3 d} \cdot T^{- c}$ .

Match exponents:

$M$ : $a + c + d = 0$

$L$ : $- 3 a + b - c - 3 d = 0$

$T$ : $- c = 1 \Rightarrow c = - 1$

From M: $a + d = 1$ . From L: $- 3 (a + d) + b - c = 0 \Rightarrow - 3 + b + 1 = 0 \Rightarrow b = 2$ .

$∴ \frac{b + c}{a + d} = \frac{2 + ( - 1 )}{1} = 1$ .

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JEE Main 2026 Jan 23 Shift 2, Physics Q23: Hydrodynamic

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