Q134 / 150 · MHT CET 2025 (25 Apr Shift 1) · Physics

The velocity of small spherical ball of mass ' $m$ ' and density ' $d_1$ ', when dropped in a container filled with glycerine becomes constant after some time. The viscous force acting on the ball if density of glycerine is ' $\mathrm{d}_2$ ' is

  • Correct answer
    A
    $\mathrm{mg}\left(1-\frac{\mathrm{d}_2}{\mathrm{~d}_1}\right)$
  • B
    $\operatorname{mg}\left(1+\frac{\mathrm{d}_2}{\mathrm{~d}_1}\right)$
  • C
    $\mathrm{mg}\left(1-\frac{\mathrm{d}_1}{\mathrm{~d}_2}\right)$
  • D
    $\mathrm{mg}\left(1+\frac{\mathrm{d}_1}{\mathrm{~d}_2}\right)$

Solution

At terminal velocity, the net force on the spherical ball must be zero, with gravitational, buoyant, and viscous forces in equilibrium. The gravitational force is $W = mg$, while the buoyant force equals the weight of displaced fluid. The volume of the sphere is $V = \frac{m}{d_1}$, so $F_B = V d_2 g = mg \frac{d_2}{d_1}$.

Balancing the forces gives $F_V + F_B = W$, which becomes $F_V + mg \frac{d_2}{d_1} = mg$. Solving for viscous force yields $F_V = mg - mg \frac{d_2}{d_1} = mg \left(1 - \frac{d_2}{d_1}\right)$.

The expression $mg \left(1 - \frac{d_2}{d_1}\right)$ corresponds to option A.

Hence, the correct answer is option: 'A'.

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