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 = Aρarbηcσd, where A is a constant and a, b, c and d are integers. The value of (b+c)/(a+d) is ______. Numerical answer type. Work it out, then open the answer.

Answer: 1. [T] = (ML⁻³)a+d Lb (ML⁻¹T⁻¹)c T: -c = 1 ⇒ c = -1; M: a + d + c = 0 ⇒ a + d = 1 L: -3(a+d) + b - c = 0 ⇒ -3 + b + 1 = 0 ⇒ b = 2 (matches t ~ ρ r²/η). (b+c)/(a+d) = 2-1/1 = 1

A ball of radius r and density ρ dropped through a viscous liquid of… | JEE Main 2026 (23 Jan, Shift 2)

Q9JEE Main 2026 · 23 JAN · SHIFT 2Medium

Applications of Dimensional Analysis

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

Numerical answer type. Work it out, then open the answer.

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Answer: 1

$[T] = (ML^{-3})^{a+d}\,L^{b}\,(ML^{-1}T^{-1})^{c}$
$T$ : $-c = 1 \Rightarrow c = -1$ ; $M$ : $a + d + c = 0 \Rightarrow a + d = 1$
$L$ : $-3(a+d) + b - c = 0 \Rightarrow -3 + b + 1 = 0 \Rightarrow b = 2$ (matches $t \sim \rho r^{2}/\eta$ ).
$\frac{b+c}{a+d} = \frac{2-1}{1} = 1$

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