In the experiment for measurement of viscosity $‘\eta ’$ of given liquid with a ball having radius R, consider following statements.

A. Graph between terminal velocity V and R will be a parabola
B. The terminal velocities of different diameter balls are constant for a given liquid
C. Measurement of terminal velocity is dependent on the temperature.
D. This experiment can be utilized to assess the density of a given liquid
E. If balls are dropped with some initial speed, the value of $\eta$ will change

Choose the correct answer from the options given below:

From Stokes' Law, the terminal velocity of a sphere moving through a viscous fluid is given by:

$$V = \frac{2}{9} \frac{R^2 (\rho_s - \rho_l) g}{\eta}$$

(A) "Graph between terminal velocity $V$ and $R$ will be a parabola."

           $$V \propto R^2$$ 

       Graphical Interpretation

• The equation $V = k R^2$ (where $k$ is a constant) is a quadratic equation in $R$.
• A quadratic equation of the form $y = kx^2$ represents a parabola.
• In this case, the relationship between $V$ and $R$ follows a parabolic curve.

         So, the graph of $V$ vs. $R$ is indeed a parabola.

(B) "The terminal velocities of different diameter balls are constant for a given liquid."

• Incorrect: Terminal velocity depends on the radius as $V \propto R^2$, so different diameter balls (different $R$) will have different terminal velocities.

(C) "Measurement of terminal velocity is dependent on the temperature."

• Correct: Viscosity $\eta$ depends on temperature; as temperature increases, $\eta$ generally decreases, causing an increase in terminal velocity. Hence, temperature affects the measurement.

(D) "This experiment can be utilized to assess the density of a given liquid."

• Correct: Since $\rho_l$ (density of liquid) appears in Stokes' Law, we can rearrange the equation to determine $\rho_l$, making it useful for assessing the density of the liquid.

(E) "If balls are dropped with some initial speed, the value of $\eta$ will change."

• Incorrect: The viscosity $\eta$ is a property of the liquid and does not change due to initial speed. The terminal velocity may take longer to reach, but $\eta$ remains constant.