Problem P. 4539. (April 2013)

P. 4539. The base of a solid 4r-high right cylinder is a circle of radius r. One side of the surface of a tale made of hard but flexible material is slippery, whilst on the other side the frictional coefficient is $\mu$ . The two parts are separated by a straight line. The cylinder is placed to the slippery side of the table (with its circular base on the tabletop), and is given a push perpendicularly to the separation line. Will the cylinder fall when it reaches the separation line or not, and if yes how, if a) $\mu$ =0.2; b) $\mu$ =0.7; c) $\mu$ =1.4? (The rotational inertia of a cylinder of mass m, radius of r and height of 4r about a horizontal axis through its centre of mass is $\frac{19}{12}mr^2$ .)

(6 pont)

Deadline expired on May 10, 2013.

Sorry, the solution is available only in Hungarian. Google translation

Megoldásvázlat. $\displaystyle a)$ Ha $\displaystyle \mu<\mu_1=\frac12$, akkor a henger nem billen fel, az alaplapján csúszva fokozatosan lefékeződik.

$\displaystyle b)$ Ha $\displaystyle \mu_1<\mu<\mu_2=\frac{31}{24}\left(\approx1{,}3\right)$, akkor a henger tömegközéppontjának mozgása fokozatosan fékeződik, s közben megemelkedik (előre billen), és ha elegendően nagy volt a kezdősebessége, felborul.

$\displaystyle c)$ Ha $\displaystyle \mu_2<\mu$, akkor a henger az érdes térfélen hirtelen lefékeződik, ,,megszorul'' (mintha egy kicsiny ütközőhöz érkezett volna), és az asztallal érintkező pontja körül elfordulva előre billen. (Ha elegendően nagy volt a kezdősebessége, a henger felborul.)

Statistics:

9 students sent a solution.

5 points: Fehér Zsombor, Sárvári Péter.

4 points: 1 student.

3 points: 2 students.

2 points: 2 students.

1 point: 2 students.

Problems in Physics of KöMaL, April 2013

9 students sent a solution.
5 points:	Fehér Zsombor, Sárvári Péter.
4 points:	1 student.
3 points:	2 students.
2 points:	2 students.
1 point:	2 students.

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