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P. 4643. Both ends of a horizontal thin tube are sealed. In the middle of the tube there is a Mercury thread of length $\displaystyle h$. The length of the air columns in both ends of the tube is $\displaystyle \ell$, and the pressure of the air in both parts is the gauge pressure of a Mercury column of height $\displaystyle H$. The tube is put into a centrifuge, which has a vertical axle, and is rotated at an angular speed of $\displaystyle \omega$.

What can the period of the oscillatory motion of the Mercury column be, provided that the temperature is constant and the oscillation is small? It can also be assumed that $\displaystyle h$ is small with respect to $\displaystyle \ell$, so the Mercury column does not break.

(5 points)

Deadline expired on 10 June 2014.

Google Translation (Sorry, the solution is published in Hungarian only.)

Megoldásvázlat. Ha $\displaystyle \omega<\omega_\text{krit.}=\sqrt{\frac{2gH}{\ell h}},$ akkor

$\displaystyle T=2\pi\sqrt{\frac{1}{\omega_\text{krit.}^2-\omega^2}},$

ha viszont $\displaystyle \omega>\omega_\text{krit.}$, úgy a rezgésidő

$\displaystyle T=2\pi \frac{\omega_\text{krit.}}{\omega}\sqrt{\frac{1}{2 ( \omega^2-\omega_\text{krit.}^2)}}.$

Statistics on problem P. 4643.
 23 students sent a solution. 5 points: Berta Dénes, Fehér Zsombor, Fekete Panna, Holczer András, Horicsányi Attila, Janzer Barnabás. 3 points: 8 students. 2 points: 6 students. 1 point: 3 students.

• Problems in Physics of KöMaL, May 2014

•  Támogatóink: Morgan Stanley