Experimental Question 1
Terminal velocity in a viscous liquid
An object falling in a liquid will eventually reach a constant velocity, called the terminal velocity. The
aim of this experiment is to measure the terminal velocities of objects falling through glycerine.
For a sphere of radius r falling at velocity v through a viscous liquid, the viscous force F is given by
F = 6πηrv. Here η is a property of the liquid called the viscosity. In this experiment you will measure
the terminal velocity of metal cylinders (because cylinders are easier to make than spheres). The diameter
of each cylinder is equal to its length, and we will assume the viscous force on such a cylinder is similar
to the viscous force on a sphere of the same diameter, 2r:
F = 6πα·ηrmv (1)
cyl
where α = 1, m = 1 for a sphere.
Preliminary
Calculation of terminal velocity (2 marks)
If ρ is the density of the cylinder and ρ0 is the density of the liquid, show that the terminal velocity vT
of the cylinder is given by
vT = Cr3−m(ρ − ρ0) (2)
where C is a constant and derive a expression for C.
Experiment
Use the equipment available to determine the numerical value of the exponent m (10 marks) and the
density of glycerine (8 marks).
Notes
† For consistency, try to ensure that the cylinders fall in the same orientation, with the axis of the
cylinder horizontal.
† The tolerances on the diameter and the length of the cylinders are 0.05 mm (you need not measure
them yourself).
† There is a brass sieve inside the container that you should use to retrieve the metal cylinders.
Important: make sure the sieve is in place before dropping objects into the glycerine, otherwise you
will not be able to retrieve them for repeat measurements.
† When glycerine absorbs water from the atmosphere, it becomes less viscous. Ensure that the
cylinder of glycerine is covered with the plastic film provided when not in use.
† Do not mix cylinders of different size and different material after the experiment.
Material Density (kgm−3)
Aluminium 2.70×103
Titanium 4.54×103
Stainless steel 7.87×103
Copper 8.96×103