Satellite A A orbits a planet at a distance d d from the planet’s center with a centripetal acceleration a0 a 0 . A second identical satellite B B orbits the same planet at a distance 2d 2 d from the planet’s center with centripetal acceleration ab a b . What is the centripetal acceleration ab a b in terms of a0 a 0 ?

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PabloSRC PabloSRC · Answer 1 · 2020-03-11T02:32:04+01:00

Answer:

[tex]a_b=\frac{1}{4}a_0[/tex]

Explanation:

Since the centripetal force F_C is, in this case, the gravitational force F_G exerted by the planet, we can say that:

[tex]F_G=F_C\\\\\frac{GMm}{r^{2} } =ma_c\\\\\frac{GM}{r^{2} }=a_c[/tex]

In the case of the satellite A we have:

[tex]\frac{GM}{d^{2} } =a_0[/tex]

And in the case of the satellite B:

[tex]\frac{GM}{(2d)^{2} } =a_b\\\\\frac{GM}{4d^{2} } =a_b\\\\\frac{GM}{d^{2} } =4a_b\\\\[/tex]

Finally, using these two expressions, we obtain :

[tex]4a_b=a_0\\\\a_b=\frac{1}{4} a_0[/tex]

In words, the centripetal acceleration a_b is equal to one fourth of a_0.