A small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho_{0}$. After sometime, the ball falls with constant velocity. The viscous force on the ball is :

(D) $\mathrm{mg}\left(1-\frac{\rho_{0}}{\rho}\right)$

(A) $\operatorname{mg}\left(\frac{\rho_{0}}{\rho}-1\right)$

(B) $\operatorname{mg}\left(1+\frac{\rho}{\rho_{0}}\right)$

(C) $\operatorname{mg}\left(1-\rho \rho_{0}\right)$

JEE Main · 2024 · 06 Apr 2024 · Shift 1 · Fluid Mechanics

A small ball of mass

m

and density

ρ

is dropped in a viscous liquid of density

ρ_{0}

. After sometime, the ball falls with constant velocity. The viscous force on the ball is :

(A) $mg (\frac{ρ _{0}}{ρ} - 1)$
(B) $mg (1 + \frac{ρ}{ρ _{0}})$
(C) $mg (1 - ρ ρ_{0})$
(D) $mg (1 - \frac{ρ _{0}}{ρ})$ ✓ Correct

Answer: (D)

Solution

At terminal (constant) velocity the net force is zero:

m g = F_{B} + F_{v}

, so

F_{v} = m g - F_{B}

.
Buoyant force

F_{B} = V ρ_{0} g = \frac{m}{ρ} ρ_{0} g

F_{v} = m g - \frac{m}{ρ} ρ_{0} g = m g (1 - \frac{ρ _{0}}{ρ}) .

JEE Main · 2024 · 06 Apr 2024 · Shift 1 · Single correct option

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