**M. 354.** Attach a spindle perpendicularly to a (preferably uniform density) rod of length \(\displaystyle h\) at a distance of \(\displaystyle \frac{1}{10}h\) from the centre of the rod. Hang the spindle horizontally with threads of lengths \(\displaystyle L\) (bifilar suspension), then make the rod swing perpendicularly to the plane determined by the threads (as shown in the *figure*).

Investigate how the two types of swinging motions (the wobbling of the rod and the waggling of the spindle) couple. Choose different thread lengths, and determine that ratio of \(\displaystyle L/h\) at which the period of the alternation between the two types of oscillation is the longest.

(6 points)

**P. 4771.** A graduated cylinder of cross-section \(\displaystyle 50~\rm cm^2\) is filled with some salt solution of density 1.2 g/cm\(\displaystyle {}^3\) up to a height of 10 cm, and then carefully (without mixing the layers) 10 cm-high clean water is stratified onto it.

\(\displaystyle a)\) What is the pressure at the bottom of the cylinder?

\(\displaystyle b)\) What is the total gravitational potential energy of the two samples of liquid with respect to the bottom of the cylinder?

By what amount will these quantities change if the two samples of liquid are mixed? (Neglect the change in the volume of the whole mixture due to the mixing process.)

(4 points)

**P. 4772.** A negligible mass and homogeneous spring spring of spring constant \(\displaystyle D\) and of length \(\displaystyle L\) is cut into \(\displaystyle n\) pieces. The small pieces of the spring are alternately attached to bodies of mass \(\displaystyle m\) each, thus creating a chain (at its end there is a body of mass \(\displaystyle m\)). Then the chain is hung.

\(\displaystyle a)\) How should the spring be cut, in order to gain the same lengths of springs in the chain when it is hanging at rest?

\(\displaystyle b)\) What distance does the centre of gravity move down if the bottom of the chain is pulled down by a distance of \(\displaystyle \delta\)?

(Let for example \(\displaystyle DL=mg\) and \(\displaystyle n=5\).)

(5 points)

**P. 4773.** A uniform thin rod of length \(\displaystyle L=2\) m is suspended with a hook at the ceiling. The rod is raised into horizontal position and it is released with zero initial velocity. After passing the vertical position, when the rod encloses an angle of \(\displaystyle 30^\circ\) with the vertical, the rod leaves the hook.

\(\displaystyle a)\) What is the least distance between the ground and the hook, if the rod arrives to the ground in a vertical position?

\(\displaystyle b)\) What is the maximum height of the bottom end of the rod during the motion?

(5 points)