Problem P. 5454. (January 2023)
P. 5454. After repairing the tires of cars, the wheels have to be balanced. The wheels are placed to a balancing machine which spins them at very high speed, and measures the imbalance of the wheel. Then small wheel weights are mounted to the appropriate positions in order to balance the wheel. A wheel is rotated at a constant angular acceleration from rest. At a certain moment the speed of the valve cap, which is at a distance of \(\displaystyle R=20\) cm from the axle, is \(\displaystyle v=1\) m/s, and its acceleration is twice as big as it was when the wheel was started to move.
\(\displaystyle a)\) How much time elapsed from the start?
\(\displaystyle b)\) What was the acceleration of the valve cap, at the start of the rotation?
\(\displaystyle c)\) How much distance did the valve cap cover during its rotation?
(4 pont)
Deadline expired on February 15, 2023.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. A szelepsapka gyorsulásvektora az érintő irányú, állandó \(\displaystyle a\) nagyságú kerületi gyorsulásból és a rá merőleges,
\(\displaystyle \frac{v^2}{R}=5\frac{\rm m}{\rm s^2}\)
centripetális gyorsulásból tevődik össze.
\(\displaystyle b)\) A megadott feltétel szerint
\(\displaystyle \sqrt{a^2+25\ \frac{\rm m^2}{\rm s^4}}=2a,\)
vagyis
\(\displaystyle a=\frac5{\sqrt3}\ \frac{\rm m }{\rm s^2}\approx 2{,}9\ \frac{\rm m }{\rm s^2}.\)
\(\displaystyle a)\) A mozgás ideje a \(\displaystyle v=at\) összefüggés szerint
\(\displaystyle t=\frac{v}{a}\approx 0{,}35\ \rm s.\)
\(\displaystyle c)\) A szelepsapka által megtett út (a körív hossza)
\(\displaystyle s=\frac{a}{2}t^2\approx 17\ \rm cm.\)
Statistics:
77 students sent a solution. 4 points: Arnold Lőrinc, Beke Bálint, Beke Botond, Benes András, Bernhardt Dávid, Bocor Gergely, Bodré Zalán, Bogdán Benedek, Bottyán Márton Péter, Chrobák Gergő, Csilling Dániel, Csonka Illés, Csornai-Metz Mátyás , Éger Viktória, Éliás Kristóf , Fajszi Karsa, Farkas Dorka Hanna, Fórizs Borbála, Kis Márton Tamás, Klement Tamás, Kovács Barnabás, Kovács Kristóf , Lévai Dominik Márk, Masa Barnabás, Merics Vilmos, Molnár Kristóf, Nemeskéri Dániel, Osváth Emese, Racskó Dániel, Schmercz Blanka, Sipeki Árpád, Susán Lőrinc Levente, Szabó Imre Bence, Szabó Márton, Tárnok Ede , Tóth Kolos Barnabás, Vágó Botond, Waldhauser Miklós, Zsova Levente. 3 points: 14 students. 2 points: 6 students. 1 point: 6 students. 0 point: 4 students. Unfair, not evaluated: 5 solutionss.
Problems in Physics of KöMaL, January 2023