In an experiment to determine the coefficient of viscosity of a liquid using the formual `eta=(piPr^(4))/(8l Q)` the radius of the capillary tube was measured to be 0.41 mm using an instrument of least count 0.001 cm, length l was measured usinga metre scale as 15 cm. The percentae error in the pressure difference is 0.4% and the percentage error in the volume of the liquid flow out per second Q is 0.3% what is the maximum percentage error in the coefficient of viscosity?

Explanation

To determine the maximum percentage error in the coefficient of viscosity (η) using the formula:

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

we need to analyze the contributions of the errors in the measurements involved in this equation.

### Step 1: Identify the variables and their errors

1. **Pressure difference (P)**: Percentage error = 0.4%
2. **Radius (r)**: Measured radius = 0.41 mm = 0.041 cm. The least count is 0.001 cm, which gives a relative error in radius.
3. **Length (l)**: Measured length = 15 cm. The error in length is negligible for this calculation.
4. **Volume flow rate (Q)**: Percentage error = 0.3%

### Step 2: Calculate the percentage error in radius (r)

The radius is measured as 0.41 mm = 0.041 cm. The least count is 0.001 cm.

The absolute error in radius (Δr) can be considered as the least count:

\[ \Delta r = 0.001 \text{ cm} \]

The percentage error in radius (r) can be calculated as follows:

\[ \text{Percentage error in } r = \left( \frac{\Delta r}{r} \right) \times 100 = \left( \frac{0.001}{0.041} \right) \times 100 \approx 2.44\% \]

### Step 3: Determine the total percentage error in η

Using the formula for the maximum percentage error in a product or quotient:

\[ \text{Maximum percentage error in } η = \text{Percentage error in } P + 4 \times \text{Percentage error in } r + \text{Percentage error in } l + \text{Percentage error in } Q \]

- The percentage error in P = 0.4%
- The percentage error in r = 2.44%
- The percentage error in l = 0% (since it is measured with a meter scale, we assume negligible error)
- The percentage error in Q = 0.3%

Now, substituting these values:

\[ \text{Maximum percentage error in } η = 0.4\% + 4 \times 2.44\% + 0\% + 0.3\% \]

Calculating:

\[ = 0.4\% + 9.76\% + 0.3\% \]
\[ = 10.46\% \]

### Final Answer

The maximum percentage error in the coefficient of viscosity (η) is approximately **10.46%**.

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