Design an experiment to measure the viscosity of liquids using a vertical funnel with a cylindrical reservoir of height \(h\) and a narrow flow section of diameter \(D\) and length \(L\). Making appropriate assumptions, obtain a relation for viscosity in terms of easily measurable quantities such as density and volume flow rate.

Question: Using a vertical funnel with a cylindrical reservoir of height \(h\), and a narrow flow section of diameter \(D\) and length \(L\), design an experiment to measure the viscosity of a liquid. Use the Hagen-Poiseuille equation and consider the flow to be laminar. Calculate the viscosity in terms of easily measurable quantities such as density and volume flow rate. Answer: To measure the viscosity of a liquid using the vertical funnel, follow these steps: 1. Fill the reservoir to height \(h\). 2. Allow the liquid to flow through the narrow flow section of diameter \(D\) and length \(L\). 3. Measure the time \(t\) it takes for a known volume \(V\) of liquid to pass through the narrow flow section. 4. Use the relation for viscosity: \(\eta = \dfrac{\rho g h \pi r^4 t}{8 L V}\), where \(\rho\) is the liquid density, \(g\) is the acceleration due to gravity, and \(r\) is the radius of the narrow flow section (i.e. \(r = D/2\)). With these measurements and the known values for \(L\), \(D\), and \(g\), the viscosity of the liquid can be determined.