Question

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

• A.
• B.
• C.
• D.

Answer 1

Answer: (B)

Answer 2

The forces acting on the ball as it moves through glycerine include the gravitational force ($mg$) and the viscous drag force ($F_v$). The motion is described by:

$mg - F_v - F_b = ma$

Where:

$F_b = \left( \frac{4}{3} \pi r^3 \right) g (\rho_L), \quad F_v = 6 \pi \eta r v$

Equating forces and simplifying:

$\left( \frac{4}{3} \pi r^3 \right) g (\rho - \rho_L) = m \frac{dv}{dt}$

Let:

$K_1 = \frac{4}{3} \pi r^3 (\rho - \rho_L) g, \quad K_2 = \frac{6 \pi \eta r}{m}$

The differential equation becomes:

$\frac{dv}{dt} = K_1 - K_2 v$

Integrating:

$\int_0^v \frac{dv}{K_1 - K_2 v} = \int_0^t dt$

Solution yields:

$v = \frac{K_1}{K_2} \left( 1 - e^{-K_2 t} \right)$