Show that the radius r of the orbit of a moon of a given planet can be determined from the radius R of the planet, the acceleration of gravity at the surface of the planet, and the time τ required by the moon to complete one full revolution about the planet. Determine the acceleration of gravity at the surface of the planet Jupiter knowing that R = 71 492 km and that τ = 3.551 days and r = 670.9 x 10³ km for its moon Europa.

g = 24.78 m/s^2

Explanation:

Given:

- R = 71492 km

- T = 3.551 days

- r = 670.9*10^3 km

Find:

- acceleration due to gravity at jupiter's surface g:

Solution:

- The time period T of a satellite in orbit is:

T = 2*pi*sqrt (r^3/GM)

GM=4*pi^2*r^3/T^2

- The local gravitational acceleration of planet is give by:

g = GM / R^2

- Combining the two expressions:

g = 4*pi^2*r^3/T^2*R^2

- Plug in values:

g = 4*pi^2*670900^3/306806^2*71492000^2

g = 24.78 m/s^2