Terminal Velocity And Stokes' Law

Question

A small steel ball falling vertically through a tall column of viscous oil eventually moves at constant speed. If a second ball of the same material but twice the radius is dropped through the identical oil, how does its terminal velocity compare?

RankGuru · Accepted Answer

Terminal velocity is reached when the downward weight of the sphere is balanced by the upward buoyant force and the viscous drag described by Stokes' law, $F = 6\pi\eta r v$. Setting net force to zero gives $\frac{4}{3}\pi r^3(\rho - \sigma)g = 6\pi\eta r v_t$, which rearranges to $v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$. For balls of the same material in the same oil, every factor except the radius is fixed, so $v_t \propto r^2$. Doubling the radius therefore multiplies the terminal velocity by $2^2 = 4$. The option 'same' ignores the radius dependence entirely. The option 'twice' assumes a linear rather than quadratic relation. The option 'half' even gives the wrong direction, since a larger ball falls faster. This quadratic dependence on radius is the central result of the NCERT treatment of viscous fall, and it holds only while the motion stays laminar so that Stokes' linear drag law applies. The weight of the sphere grows as $r^3$ whereas the viscous drag grows only as $r$, so their balance forces the steady speed to climb as $r^2$. As a check, larger raindrops and hailstones are known to reach higher steady speeds than tiny droplets, which agrees with the strong $r^2$ growth obtained here.

Terminal Velocity And Stokes' Law

Select the correct option:

🔒 Solution Hidden from View

More properties of solids and liquids Practice Questions

If the absolute temperature of a perfectly black body radiating heat is doubled, by what factor does...

When a substance is melting at its melting point while heat is continuously supplied, what happens t...

In a hydraulic lift the small piston has an area of 0.01 m² and the large piston has an area of 0.5 ...

How much heat energy must be supplied to raise the temperature of 0.5 kg of water from 25°C to 75°C,...

A metal rod of length 1 m at 20°C has a coefficient of linear expansion of 2 \times 10^{-5} per °C; ...

The Reynolds number is used to predict the nature of fluid flow in a pipe; a very high Reynolds numb...

A hot body cools from 80°C to 70°C in 5 minutes when placed in surroundings at 30°C; according to Ne...

Two metal rods of identical length and cross-section but different thermal conductivities are joined...

A wire is stretched so that it stores elastic potential energy; if the same wire is stretched to twi...

A diver descends to a depth of 20 m below the surface of a lake where water density is 1000 kg/m³; w...

During winter a lake freezes only at its surface while fish survive in the water below; this is best...

A steel wire of length 2 m and cross-sectional area 1 mm² is stretched by a load of 100 N, producing...

About This Question

Solution