In the experiment for measurement of viscosity \( \eta \) of a 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.
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.
Show Hint
- \(B\), \(D\) and \(E\) Only
- \(A\), \(C\) and \(D\) Only
- \(A\), \(B\) and \(E\) Only
- \(C\), \(D\) and \(E\) Only
The Correct Option is B
Approach Solution - 1
Let's analyze each statement:
A. The terminal velocity $ V $ is given by:
$$ V = \frac{2}{9} \frac{r^2 (\rho_s - \rho_l)g}{\eta} $$
where $ r $ is the radius of the ball, $ \rho_s $ is the density of the ball, $ \rho_l $ is the density of the liquid, $ g $ is the acceleration due to gravity, and $ \eta $ is the viscosity of the liquid.
Since $ V \propto r^2 $, the graph between terminal velocity $ V $ and $ r $ (or $ R $) will be a parabola.
So, statement A is correct.
B. The terminal velocity depends on the radius (or diameter) of the ball. Different diameter balls will have different terminal velocities for a given liquid.
So, statement B is incorrect.
C. Measurement of terminal velocity is dependent on the temperature. The viscosity of the liquid is temperature-dependent. As temperature increases, the viscosity of most liquids decreases, affecting the terminal velocity.
So, statement C is correct.
D. This experiment can be utilized to assess the density of a given liquid. By measuring the terminal velocity of a ball with known density and radius, and knowing the viscosity, we can solve for the density of the liquid in the equation for terminal velocity.
So, statement D is correct.
E. If balls are dropped with some initial speed, the value of $ \eta $ will not change. The viscosity is a property of the liquid and does not depend on the initial speed of the ball. Although the ball takes a longer/shorter amount of time depending on whether it is thrown or released, this does not affect the viscosity.
So, statement E is incorrect.
Conclusion:
The correct statements are A, C, and D.
Final Answer:
The final answer is $ (2)\ \text{A, C and D only} $.
Approach Solution -2
The experiment is based on measuring the viscosity \( \eta \) of a given liquid using Stokes’ law. When a spherical ball of radius \( R \) falls through a viscous liquid, it eventually attains a constant speed called the terminal velocity \( V \). The relationship between these parameters is given by:
\[ V = \frac{2}{9} \frac{( \rho_s - \rho_l ) g R^2}{\eta} \] where \( \rho_s \) is the density of the sphere, \( \rho_l \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( \eta \) is the viscosity of the liquid.
Step 2: Analyze each statement.
(A) Graph between terminal velocity \( V \) and \( R \):
From the above equation, \( V \propto R^2 \). Hence, a plot of \( V \) vs \( R \) will be a parabola.
✅ Statement (A) is correct.
(B) The terminal velocities of different diameter balls are constant for a given liquid:
Since \( V \) depends on \( R^2 \), terminal velocity varies with radius. Thus, different diameters give different velocities.
❌ Statement (B) is incorrect.
(C) Measurement of terminal velocity is dependent on temperature:
Viscosity \( \eta \) depends on temperature — as temperature increases, viscosity decreases. Hence, terminal velocity changes with temperature.
✅ Statement (C) is correct.
(D) The experiment can be used to assess the density of a liquid:
From Stokes’ law, if \( \eta \) is known, the density of the liquid \( \rho_l \) can be determined by measuring \( V \), \( R \), and \( \rho_s \).
✅ Statement (D) is correct.
(E) If balls are dropped with some initial speed, the value of \( \eta \) will change:
Even if the balls are dropped with an initial speed, they soon attain terminal velocity due to viscous drag, and the viscosity \( \eta \) (a property of the liquid) remains constant.
❌ Statement (E) is incorrect.
Step 3: Conclusion.
The correct statements are \( A \), \( C \), and \( D \) only.
Final Answer:
\[ \boxed{A, C \text{ and } D \text{ only}} \]
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