A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the acceleration of the satellite is equal to g(R/r)², where g is the acceleration of gravity at the surface of the planet, R is the radius of the planet, and r is the distance from the center of

Question

A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the acceleration of the satellite is equal to g(R/r)², where $g$ is the acceleration of gravity at the surface of the planet, R is the radius of the planet, and r is the distance from the center of the planet to the satellite. Knowing that the diameter of the sun is 1.39 Gm and that the acceleration of gravity at its surface is 274 m/s², determine the radius of the orbit of the indicated planet around the sun assuming that the orbit is circular.

Earth: ${ \left( v_{ mean } \right) }_{ orbit }=107 \ Mm/h$ .

Earth:[latex]{ \left( v_{ mean } \right) }_{ orbit }=107 \ Mm/h[/latex].

Accepted Answer

For the sun,

[latex]g=274 \mathrm{~m} / \mathrm{s}^{2}[/latex]

and

[latex]R=\frac{1}{2} D=\left\lgroup\frac{1}{2} ight group\left(1.39 imes 10^{9} ight)=0.695 imes 10^{9} \mathrm{~m}[/latex]

Given that [latex]a_{n}=\frac{g R^{2}}{r^{2}}[/latex] and that for a circular orbit [latex]a_{n}=\frac{ u^{2}}{r}[/latex]

Eliminating [latex]a_{n}[/latex] and solving for r,

[latex]r=\frac{g R^{2}}{ u^{2}}[/latex]

For the planet Earth,

[latex] u=107 imes 10^{6} \mathrm{~m} / \mathrm{h}=29.72 imes 10^{3} \mathrm{~m} / \mathrm{s}[/latex]

Then

[latex]r=\frac{(274)\left(0.695 imes 10^{9} ight)^{2}}{\left(29.72 imes 10^{3} ight)^{2}}=149.8 imes 10^{9} \mathrm{~m}[/latex]

[latex]r=149.8 \mathrm{\ Gm}\blacktriangleleft[/latex]

Question 11.53: A satellite will travel indefinitely in a circular orbit aro...

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