Problem P. 5115. (March 2019)
P. 5115. The mass of an exoplanet, whose mass distribution has a spherical symmetry, is four times that of the Earth, and the acceleration due to gravity on the surface of the – non-rotating – planet is twice of the gravitational acceleration on the Earth.
\(\displaystyle a)\) What is the radius of the exoplanet, and what is its average density?
\(\displaystyle b)\) At what speed should an object be projected in order that it undergoes uniform circular motion right above the surface of the exoplanet?
(5 pont)
Deadline expired on April 10, 2019.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. \(\displaystyle a)\) A nehézségi gyorsulás \(\displaystyle (g)\) a bolygó talajszintjén \(\displaystyle M/R^2\)-tel arányos, így
\(\displaystyle \frac{g_\text{bolygó}}{g_\text{Föld}}=\frac{M_\text{bolygó}}{M_\text{Föld}}\left(\frac{R_\text{Föld}}{R_\text{bolygó}}\right)^2=2,\)
ahonnan
\(\displaystyle \frac{R_\text{bolygó}}{R_\text{Föld}}=\sqrt{2},\qquad R_\text{bolygó}\approx 9010~\rm km.\)
Az átlagsűrűség \(\displaystyle M/R^3\)-nel arányos, így
\(\displaystyle \frac{\varrho_\text{bolygó}}{\varrho_\text{Föld}}=\frac{M_\text{bolygó}}{M_\text{Föld}}\left(\frac{R_\text{Föld}}{R_\text{bolygó}}\right)^3=\sqrt2, \qquad {\varrho_\text{bolygó}}\approx \sqrt{2}\cdot 5{,}5~\frac{\rm kg}{\rm dm^3}\approx7{,}8~\frac{\rm kg}{\rm dm^3}.\)
\(\displaystyle b)\) Az első kozmikus sebesség (a bolygó felületének közelében) \(\displaystyle \sqrt{M/R}\)-rel arányos, így
\(\displaystyle v^{(I)}_\text{bolygó}=\sqrt[4]{8}\cdot v^{(I)}_\text{Föld}\approx 13{,}3~\frac{\rm km}{\rm s}.\)
Statistics:
72 students sent a solution. 5 points: 51 students. 4 points: 14 students. 3 points: 5 students. 2 points: 2 students.
Problems in Physics of KöMaL, March 2019