To study the laminar flow of a liquid through a narrow tube, the following formula is used `V=(pi P r^(4))/(8 eta l)`. Given P = 76 cm of Hg , r = 0.28 cm , V = 1.2 `cm^(2)s^(-1)` , l = 18.2 cm
If these quantities are measured to the accuracy of 0.5 cm of Hg, `0.01 cm, 0.1 cm^(3) s^(-1)` and 0.1 cm respectively, find maximum percentatge error in the value of `eta`.

Options

A. `18.83%`
B. `23.8 %`
C. `38.83%`
D. `48.83%`

Answer

To find the maximum percentage error in the value of \( \eta \) (viscosity) using the formula

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

we first rearrange the formula to express \( \eta \):

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

### Step 1: Identify the variables and their errors
Given:
- \( P = 76 \, \text{cm of Hg} \) with an error \( \Delta P = 0.5 \, \text{cm of Hg} \)
- \( r = 0.28 \, \text{cm} \) with an error \( \Delta r = 0.01 \, \text{cm} \)
- \( V = 1.2 \, \text{cm}^2/\text{s} \) with an error \( \Delta V = 0.1 \, \text{cm}^2/\text{s} \)
- \( l = 18.2 \, \text{cm} \) with an error \( \Delta l = 0.1 \, \text{cm} \)

### Step 2: Calculate the relative errors
The formula for the relative error in \( \eta \) is given by:

\[ \frac{\Delta \eta}{\eta} \times 100 = \frac{\Delta P}{P} \times 100 + 4 \frac{\Delta r}{r} \times 100 + \frac{\Delta V}{V} \times 100 + \frac{\Delta l}{l} \times 100 \]

### Step 3: Substitute the values
1. Calculate \( \frac{\Delta P}{P} \):
\[ \frac{\Delta P}{P} = \frac{0.5}{76} \approx 0.00658 \quad \Rightarrow \quad \frac{\Delta P}{P} \times 100 \approx 0.658\% \]

2. Calculate \( 4 \frac{\Delta r}{r} \):
\[ \frac{\Delta r}{r} = \frac{0.01}{0.28} \approx 0.03571 \quad \Rightarrow \quad 4 \frac{\Delta r}{r} \times 100 \approx 14.29\% \]

3. Calculate \( \frac{\Delta V}{V} \):
\[ \frac{\Delta V}{V} = \frac{0.1}{1.2} \approx 0.08333 \quad \Rightarrow \quad \frac{\Delta V}{V} \times 100 \approx 8.33\% \]

4. Calculate \( \frac{\Delta l}{l} \):
\[ \frac{\Delta l}{l} = \frac{0.1}{18.2} \approx 0.00549 \quad \Rightarrow \quad \frac{\Delta l}{l} \times 100 \approx 0.55\% \]

### Step 4: Sum the percentage errors
Now, we add all the percentage errors together:
\[ \text{Total Percentage Error} = 0.658 + 14.29 + 8.33 + 0.55 \approx 23.83\% \]

### Final Step: Conclusion
Thus, the maximum percentage error in the value of \( \eta \) is approximately:

\[ \boxed{23.83\%} \]