Inspired by this answer to a question on Puzzling.SE (which contains the old but wrong idea that the Coriolis force determines the direction in which a bathroom vortex rotates, and also an approach using a very short Foucault pendulum), is there a practical experiment that can be performed with household items in a closed room to determine with some accuracy whether one is currently located on the northern or southern hemisphere of the Earth? Assume the room is 2.5 meters, not 15 meters high like most Foucault installations.
Method 1: Foucault Pendulum
As user Rob asks, what is wrong with a short Foucault pendulum?
There is a problem, but it can be overcome inexpensively, but with some DIY effort at home. The problem is that, by dint of imperfections in the suspending fibre and bob, no pendulum will swing in a plane even if the Earth were not rotating. Instead, the bob will make an elliptical path each cycle (think of this as two vibration eigenmodes in orthogonal planes being inevitably excited by imperfections and asymmetries). As soon as there is any elliptical as opposed to straight planar motion, the ellipse precesses at a rate proportional to its minor axis. The minor axis is always nonzero owing to imperfections and asymmetries and one cannot eliminate these altogether. Therefore, a short putative Foucault pendulum will precess at a rate set wholly by its imperfections, not by the Earth's rotation. The latter is overwhelmed.
The two papers below propose and demonstrate a simple feedback circuit to impose a counteracting force to null out the elliptical motion and keep the motion planar. Using this feedback circuit, a Foucault pendulum of three metres length whose precession is dominated by the Earth's rotation can be built. Experimental results are presented showing that the device can clearly measure the Earth's rotation.
Method 2: Magnetic Dip
Build your own Dip Circle and measure the Magnetic Dip (see Wiki Page of this name) in your room. With a compass, you know which way North is, so you can tell the sign of the dip. Dip is negative in the Southern Hemisphere, and positive in the Northern.