Question

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

Answer 1

To solve this problem it is necessary to use the concepts related to the Gravitational Force and Newton's Second Law, as far as we know:

$F_g = \frac{GMm}{r^2}$

Where,

G = Gravitational constant

M = Mass of earth (in this case)

m = mass of satellite

r = radius

In the other hand we have the second's newton law:

$F = ma$

Where,

m = mass

a = acceleration

Equation both equations we have,

$ma = \frac{GMm}{r^2}$

For the problem we have that,

Satellite A:

$ma_A = \frac{GMm}{r^2}$

Satellite B:

$ma_B = \frac{GMm}{(2r)^2}$

The ratio between the two satellites would be,

$\frac{ma_A}{ma_B}= \frac{\frac{GMm}{r^2}}{\frac{GMm}{(2r)^2}}$

Solving for a_B,

$a_B = \frac{a_A}{4}$

Therefore the centripetal acceleration of  $A_B$ is a quarter of $a_A$