The coefficient of viscosity ($\eta$) of a liquid by the method of flow through a capillary tube is given by the formula 

$\eta =\frac{\mathrm{\pi }}{8}\frac{R^4}{l}\frac{P}{Q}$

where R = radius of the capillary tube, 

          l = length of the tube, 

         P = pressure difference between its ends, 

         and 

        Q = volume of liquid flowing per second. 

Which quantity must be measured most accurately?

The coefficient of viscosity (η) of a liquid by the method of flow through a capillary tube is given by the formula:

η = (πR⁴P) / (8lQ)

Explanation of the Variables:

• R = radius of the capillary tube
• l = length of the tube
• P = pressure difference between the ends of the tube
• Q = volume of liquid flowing per second

Importance of Accurate Measurement:

Let's analyze each variable to determine which needs to be measured most accurately:

1. R (Radius of the Capillary Tube):

The radius appears to the fourth power (R⁴), meaning even small errors in measuring R can lead to significant changes in the value of viscosity (η). A small percentage error in R results in a much larger error in η.

2. l (Length of the Tube):

The length of the tube (l) is directly proportional to η, but it does not have the same impact as R. A relatively large error in measuring l will not result in as large a change in η as a similar error in R.

3. P (Pressure Difference):

The pressure difference (P) affects η directly and linearly. Any error in measuring P will affect the value of viscosity, but not as dramatically as errors in R.

4. Q (Volume of Liquid Flowing per Second):

The volume of liquid flowing per second (Q) is in the denominator of the formula. However, it is less sensitive than R in terms of its impact on the viscosity value.

Final Answer

Since R is raised to the fourth power, even small errors in measuring the radius R will cause significant errors in the calculated viscosity.