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

The Correct Option is B

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:

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