In determing viscosity `(eta)` by poiseuille's method for formula used `eta = (pi p r^4)/(8 vl)`. Which of the quantities in the formula must be measured more accurately

Options

A. p
B. r
C. v
D. l

Answer

To determine which quantity in the formula for viscosity \( \eta \) must be measured more accurately, we can analyze the formula given:

\[ \eta = \frac{\pi P r^4}{8 V L} \]

where:
- \( P \) is the pressure,
- \( r \) is the radius of the tube,
- \( V \) is the volume flow rate,
- \( L \) is the length of the tube.

### Step-by-Step Solution:

1. **Identify the formula components**:
The formula consists of several variables: pressure \( P \), radius \( r \), volume flow rate \( V \), and length \( L \).

2. **Understand the powers of each variable**:
In the formula, the powers of the variables are as follows:
- \( P \) has a power of 1,
- \( r \) has a power of 4,
- \( V \) has a power of 1,
- \( L \) has a power of 1.

3. **Calculate the relative error contribution**:
The relative error in \( \eta \) can be expressed as:
\[ \frac{\Delta \eta}{\eta} = \frac{\Delta P}{P} + 4 \frac{\Delta r}{r} + \frac{\Delta V}{V} + \frac{\Delta L}{L} \]
Here, \( \Delta \) represents the uncertainty in the respective measurements.

4. **Analyze the relative error contributions**:
- The term \( \frac{\Delta P}{P} \) contributes linearly (1 times),
- The term \( 4 \frac{\Delta r}{r} \) contributes four times,
- The terms \( \frac{\Delta V}{V} \) and \( \frac{\Delta L}{L} \) also contribute linearly (1 times each).

5. **Determine the dominant term**:
Since the radius \( r \) has a power of 4, any small error in measuring \( r \) will have a much larger effect on the calculated viscosity \( \eta \) compared to the other variables. This means that the measurement of \( r \) is the most critical.

6. **Conclusion**:
Therefore, the quantity that must be measured more accurately in the viscosity formula is the radius \( r \).

### Final Answer:
The quantity that must be measured more accurately is the radius \( r \).

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