Problem G. 760. (November 2021)

G. 760. A 10 cm high aluminium cone is raised slowly out of an aquarium, by means of a thread attached to the apex of the cone. The shape of the aquarium is a rectangular box, and the diameter of the cone is also 10 cm. Initially the base of the cone lies on the bottom of the aquarium, and the cone is totally submerged into the water. The volume of the aquarium is much greater than that of the cone.

Plot the graph of the tension in the thread as a function of the displacement of the cone.

(4 pont)

Deadline expired on December 15, 2021.

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

Megoldás. Az \(\displaystyle r\) alapkör-sugarú, \(\displaystyle h\) magas kúp térfogata

\(\displaystyle V_0=\frac{1}{3}r^2\pi h=0{,}26~\rm dm^3,\)

súlya levegőben

\(\displaystyle G_0=V_0\varrho_\text{Al}=6{,}9~\rm N,\)

vízben pedig

\(\displaystyle G_1=G_0\left(1-\frac{\varrho_\text{víz}}{\varrho_\text{Al}}\right)=4{,}3~\rm N.\)

Jelöljük a kúp elmozdulását \(\displaystyle x\)-szel, és legyen \(\displaystyle x=0\) az a helyzet, amikor a kúp csúcsa éppen a vízfelszínnél van. (Ha kezdetben a víz \(\displaystyle H\) magasan ellepte a kúpot, akkor \(\displaystyle x\ge -H\).) A fonalat feszítő erő \(\displaystyle x\) függvényében

\(\displaystyle G(x)\equiv G_1, \qquad \text{ha}~-H<x<0,\)

\(\displaystyle G(x)\equiv G_0, \qquad \text{ha}~ x>h,\)

és

\(\displaystyle G(x)\equiv G_1+\frac{x^3}{h^3}\left(G_0-G_1\right).\)

Kihasználtuk, hogy az eredeti kúphoz hasonló, de csak \(\displaystyle x\) magasságú kúp térfogata \(\displaystyle \frac{x^3}{h^3}V_0.\)

Statistics:

25 students sent a solution.
4 points:	Beke Botond, Bocor Gergely, Hruby Laura, Kovács Klára, Sütő Áron.
3 points:	Heisz András Botond, Jacsman Vencel , Kiss 668 Benedek, Kiss 987 Barnabás, Marosi Botond Máté, Medgyesi Júlia, Richlik Márton, Sós Ádám, Zsova Levente.
2 points:	1 student.
1 point:	4 students.
0 point:	2 students.
Unfair, not evaluated:	1 solutions.
Not shown because of missing birth date or parental permission:	1 solutions.

Problems in Physics of KöMaL, November 2021