matematikk.net

Et sted mellom jorda og et andre legeme, vil et tredje legeme bli påvirket av like store gravitasjonskrefter fra de to.

Kall det andre legemet [tex]\alpha[/tex]

Kall det tredje legemet [tex]\delta[/tex]

Avstanden fra jord til legemet er gitt ved [tex]z[/tex] og avstanden fra jordsentrum til det andre legemet er gitt ved [tex]r[/tex], forutsett at du kjenner [tex]r[/tex].

Forklar og bevis at det tredje legemets-posisjon (avstand fra jordsentrum) kan gis ved [tex]z=\frac{r\sqrt{\frac{m_j}{m_\alpha}}}{\sqrt{\frac{m_j}{m_\alpha}}+1}[/tex]

Hvor [tex]m_j[/tex] er jordens masse og [tex]m_\alpha[/tex] er andre legemets masse.

Edit: Liten forandring

$F_j = G \frac{m_{\delta} m_{j}}{r_{j,\delta}^2}$

$F_{\alpha} = G \frac{m_{\delta} m_{\alpha}}{r_{\alpha, \delta}^2}$

$F_j = F_{\alpha}$

$G \frac{m_{\delta} m_{j}}{r_{j,\delta}^2} = G \frac{m_{\delta} m_{\alpha}}{r_{\alpha, \delta}^2}$
$\frac{r_{j, \delta}^2}{m_j} = \frac{r_{\alpha, \delta}^2}{m_{\alpha}}$

$r_{j, \delta} = r_{\alpha, \delta} \sqrt{\frac{m_j}{m_{\alpha}}}$

$r_{\alpha, \delta} + r_{j, \delta} = r_{j, \alpha}$

$r_{j, \delta} = (r_{j, \alpha}-r_{j, \delta}) \sqrt{\frac{m_j}{m_{\alpha}}}$
$r_{j, \delta}+r_{j, \delta}\sqrt{\frac{m_j}{m_{\alpha}}} = r_{j, \alpha} \sqrt{\frac{m_j}{m_{\alpha}}}$
$r_{j, \delta} (1+\sqrt{\frac{m_j}{m_{\delta}}})=r_{j, \alpha} \sqrt{\frac{m_j}{m_{\alpha}}}$
$r_{j, \delta} = \dfrac{r_{j, \alpha} \sqrt{\frac{m_j}{m_{\alpha}}}}{\sqrt{\frac{m_j}{m_{\alpha}}}+1}$

Gjest skrev:$F_j = G \frac{m_{\delta} m_{j}}{r_{j,\delta}^2}$

$F_{\alpha} = G \frac{m_{\delta} m_{\alpha}}{r_{\alpha, \delta}^2}$

$F_j = F_{\alpha}$

$G \frac{m_{\delta} m_{j}}{r_{j,\delta}^2} = G \frac{m_{\delta} m_{\alpha}}{r_{\alpha, \delta}^2}$
$\frac{r_{j, \delta}^2}{m_j} = \frac{r_{\alpha, \delta}^2}{m_{\alpha}}$

$r_{j, \delta} = r_{\alpha, \delta} \sqrt{\frac{m_j}{m_{\alpha}}}$

$r_{\alpha, \delta} + r_{j, \delta} = r_{j, \alpha}$

$r_{j, \delta} = (r_{j, \alpha}-r_{j, \delta}) \sqrt{\frac{m_j}{m_{\alpha}}}$
$r_{j, \delta}+r_{j, \delta}\sqrt{\frac{m_j}{m_{\alpha}}} = r_{j, \alpha} \sqrt{\frac{m_j}{m_{\alpha}}}$
$r_{j, \delta} (1+\sqrt{\frac{m_j}{m_{\delta}}})=r_{j, \alpha} \sqrt{\frac{m_j}{m_{\alpha}}}$
$r_{j, \delta} = \dfrac{r_{j, \alpha} \sqrt{\frac{m_j}{m_{\alpha}}}}{\sqrt{\frac{m_j}{m_{\alpha}}}+1}$

Fint!