Physics - Fluid Question with Solution | TestHub

When a small metal sphere falls through a viscous liquid, the forces acting on it are its weight ($W$) downwards, buoyant force ($F_B$) upwards, and viscous drag force ($F_v$) upwards.

The equation of motion is given by:
$m\frac{dv}{dt}=W-F_B-F_v$
$m\frac{dv}{dt}=mg-\frac{4}{3}\pi r^3\sigma g-6\pi \eta rv$

Initially at $t=0$, $v=0$, so the acceleration is maximum. As velocity increases, the viscous drag force ($6\pi \eta rv$) increases, which decreases the net force and hence the acceleration.

Eventually, the net force becomes zero, and the sphere attains a constant maximum velocity called the terminal velocity ($v_T$).

The velocity varies with time according to the relation:
$v=v_T(1-e^{-kt/m})$

This represents an exponentially increasing curve that starts from the origin and asymptotically approaches a constant value. Graph (d) correctly represents this variation.

Answer: d