A spherical body of radius $r$ and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $\mathrm{v}_{0}$. Estimated maximum error in the quantity $\eta$ is : (Ignore errors associated with $\sigma, \quad \rho$ and g , gravitational acceleration)

(B) $\frac{2 \Delta r}{r}+\frac{\Delta v_{0}}{v_{0}}$

(A) $2 \frac{\Delta r}{r}-\frac{\Delta v_{0}}{v_{0}}$

(C) $2\left[\frac{\Delta \mathrm{r}}{\mathrm{r}}+\frac{\Delta \mathrm{v}_{0}}{\mathrm{v}_{0}}\right]$

(D) $2\left[\frac{\Delta \mathrm{r}}{\mathrm{r}}-\frac{\Delta \mathrm{v}_{0}}{\mathrm{v}_{0}}\right]$

JEE Main · 2026 · 21 Jan 2026 · Shift 2 · Units & Measurement

A spherical body of radius

r

and density

σ

falls freely through a viscous liquid having density

ρ

and viscosity

η

and attains a terminal velocity

v_{0}

. Estimated maximum error in the quantity

η

is : (Ignore errors associated with

σ, ρ

and g , gravitational acceleration)

Answer: (B)

Terminal velocity is given by

v_{0} = \frac{2 r ^{2} g ( σ - ρ )}{9 η}

. Rearranging:

η = \frac{2 r ^{2} g ( σ - ρ )}{9 v _{0}}

. Taking logarithmic differentiation (ignoring errors in

σ, ρ, g

\frac{Δ η}{η} = 2 \frac{Δ r}{r} + \frac{Δ v _{0}}{v _{0}}

JEE Main · 2026 · 21 Jan 2026 · Shift 2 · Single correct option