Problem P. 4641. (May 2014)

P. 4641. An inclined plane of angle of elevation \(\displaystyle \alpha\) is placed to a horizontal surface, and then an object of mass \(\displaystyle m\) is placed to the top of the inclined plane. The small mass can slide on the slope without friction. With the help of the horizontal plane it is assured that the slope accelerates in the vertical direction at an acceleration of \(\displaystyle a<g\).

\(\displaystyle a)\) Determine the horizontal and the vertical components of the acceleration of the object sliding down the slope, and the equation of the path of the sliding object. (The object is always in contact with the slope.)

\(\displaystyle b)\) How long does it take for the object to reach the bottom of the slope, if the length of the slope is \(\displaystyle L\)?

\(\displaystyle c)\) What is the force exerted by the object on the slope?

Data: \(\displaystyle \alpha=30^\circ\), \(\displaystyle L=1.5\) m, \(\displaystyle m=4\) kg, \(\displaystyle a=6~\rm m/s^2\).

(4 pont)

Deadline expired on June 10, 2014.

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

Megoldásvázlat. \(\displaystyle a)\) A vízszintes irányú gyorsulás:

\(\displaystyle a_x=(g-a)\sin\alpha\,\cos\alpha=1{,}65~\rm m/s^2 ,\)

a függőleges gyorsulás (a pozitív irány felfelé választva)

\(\displaystyle a_y=-g\sin^2\alpha-a\cos^2\alpha=-6{,}95~\rm m/s^2.\)

A pálya egyenlete (az origót a lejtő legmagasabb pontjához helyezve):

\(\displaystyle y(x)=-\frac{g\,{\rm tg}\alpha+a\,{\rm ctg}\alpha}{g-a}\,x=-4{,}21\,x.\)

\(\displaystyle b)\) \(\displaystyle t=\sqrt{\frac{2L}{(g-a)\sin\alpha}}=1{,}25~\rm s.\)

\(\displaystyle c)\) \(\displaystyle N=m(g-a)\cos\alpha=13{,}2~\rm N.\)

Statistics:

62 students sent a solution.
4 points:	Balogh Menyhért, Berta Dénes, Büki Máté, Csathó Botond, Di Giovanni Márk, Eper Miklós, Farkas Tamás, Fekete Panna, Holczer András, Horicsányi Attila, Janzer Barnabás, Juhász Péter, Morvay Bálint, Nagy Zsolt, Németh Flóra Boróka, Olosz Balázs, Öreg Botond, Pázmán Zalán, Radnóti Réka, Sal Kristóf, Seress Dániel, Szántó Benedek, Szász Norbert Csaba, Takács Péter György, Varju Ákos, Wiandt Péter.
3 points:	Antalicz Balázs, Blum Balázs, Forrai Botond, Gróf Tamás, Jakus Balázs István, Kaposvári Péter, Kovács Péter Tamás, Molnár Ádám, Rózsa Tibor, Szentivánszki Soma , Tanner Martin.
2 points:	11 students.
1 point:	11 students.
0 point:	3 students.

Problems in Physics of KöMaL, May 2014