Question

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 ?

Answer 1

Answer:

$a_b=\frac{1}{4}a_0$

Explanation:

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

$F_G=F_C\\\\\frac{GMm}{r^{2} } =ma_c\\\\\frac{GM}{r^{2} }=a_c$

In the case of the satellite A we have:

$\frac{GM}{d^{2} } =a_0$

And in the case of the satellite B:

$\frac{GM}{(2d)^{2} } =a_b\\\\\frac{GM}{4d^{2} } =a_b\\\\\frac{GM}{d^{2} } =4a_b$

Finally, using these two expressions, we obtain :

$4a_b=a_0\\\\a_b=\frac{1}{4} a_0$

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