When an object moves through a fluid, as when a ball falls through air or a glass sphere falls through water te fluid exerts a viscous foce F on the object this force tends to slow the object for a small sphere of radius `r` moving is given by stoke's law, `F_(w)=6pietarv`. in this formula `eta` in the coefficient of viscosity of the fluid which is the proportionality constant that determines how much tangential force is required to move a fluid layer at a constant speed v, when the layer has an area A and is located a perpendicular distance z from and immobile surface. the magnitude of the force is given by `F=etaAv//z`. For a viscous fluid to move from location 2 to location 1 along 2 must exceed that at location 1, poiseuilles's law given the volumes flow rate Q that results from such a pressure difference `P_(2)-P_(1)`. The flow rate of expressed by the formula `Q=(piR^(4)(P_(2)-P_(1)))/(8etaL)` poiseuille's law remains valid as long as the fluid flow is laminar. For a sfficiently high speed however the flow becomes turbulent flow is laminar as long as the reynolds number is less than approximately 2000. This number is given by the formula `R_(e)=(2overline(v)rhoR)/(eta)` In which `overline(v)` is the average speed `rho` is the density `eta` is the coefficient of viscosity of the fluid and R is the radius of the pipe. Take the density of water to be `rho=1000kg//m^(3)`
Q. Which of the following may be concluded from the information in the passage?

Options

A. the volume flow rate and the mass flow rate are the more for more viscous fluids.
B. the volume flow rate is smaller for more viscous fluids whereas the mass flow rate is greater for more viscous fluids.
C. The volume flow rate is greater for more viscous fluids whereas the mass flow rate is smaller for more viscous fluids.
D. The volume flow rate and the mass flow rate are greater for less viscous fluids.

Answer

To solve the question based on the provided information, we will analyze the relationships between the volume flow rate, mass flow rate, and the coefficient of viscosity of fluids.

### Step-by-Step Solution:

1. **Understanding Viscous Force**:
- According to Stokes' law, the viscous force \( F_w \) acting on a small sphere moving through a fluid is given by:
\[ F_w = 6 \pi \eta r v \]
- Here, \( \eta \) is the coefficient of viscosity, \( r \) is the radius of the sphere, and \( v \) is the velocity of the sphere.

2. **Volume Flow Rate**:
- Poiseuille's law states that the volume flow rate \( Q \) through a pipe is given by:
\[ Q = \frac{\pi R^4 (P_2 - P_1)}{8 \eta L} \]
- From this equation, we can see that the volume flow rate \( Q \) is inversely proportional to the coefficient of viscosity \( \eta \). This means:
\[ Q \propto \frac{1}{\eta} \]
- Therefore, as the viscosity increases, the volume flow rate decreases.

3. **Mass Flow Rate**:
- The mass flow rate \( \dot{m} \) can be related to the volume flow rate by the equation:
\[ \dot{m} = \rho Q \]
- Since \( Q \) is inversely proportional to \( \eta \), it follows that:
\[ \dot{m} \propto \frac{1}{\eta} \]
- This indicates that the mass flow rate also decreases as the viscosity increases.

4. **Conclusion**:
- Both the volume flow rate \( Q \) and the mass flow rate \( \dot{m} \) are inversely proportional to the coefficient of viscosity \( \eta \). Therefore, for less viscous fluids (lower \( \eta \)), both the volume flow rate and mass flow rate will be higher.

5. **Analyzing the Options**:
- Based on the conclusions drawn:
- Option 1: Incorrect - states that both rates are more for more viscous fluids.
- Option 2: Incorrect - states volume flow rate is smaller for more viscous fluids but incorrectly claims mass flow rate is greater.
- Option 3: Incorrect - states volume flow rate is greater for more viscous fluids.
- Option 4: Correct - states that volume flow rate and mass flow rate are greater for less viscous fluids.

### Final Answer:
The correct conclusion from the information in the passage is **Option 4**: Volume flow rate and mass flow rate are greater for less viscous fluids.