Question

In order for a satellite to move in a stable circular orbit of radius 6588km at a constant speed, its centripetal acceleration must be inversely proportional to the square of the radius r of the orbit. What is the speed of the satellite and the time required to complete one orbit? The universal gravitational constant is 6.67259e-11 Nm^2/kg^2 and the mass of the earth is 5.98e24kg.

Answer 1

Explanation:

Given that,

Radius in which the satellite orbits, r = 6588 km

Solution,

The centripetal force acting on the satellite is balanced by the gravitational force acting between earth and the satellite. Its expression can be written by :

$\dfrac{GmM}{r^2}=\dfrac{mv^2}{r}$

$v=\sqrt{\dfrac{GM}{r}}$, M is the mass of earth

$v=\sqrt{\dfrac{6.67259\times 10^{-11}\times 5.98\times 10^{24}}{6588\times 10^3}}$

v = 7782.53 m/s

Let t is the time required to complete one orbit. It can be calculated as :

$t=\dfrac{d}{v}$

$t=\dfrac{2\pi r}{v}$

$t=\dfrac{2\pi \times 6588\times 10^3}{7782.53}$

t = 5318.78 seconds

or

t = 1.47 hour

Therefore, this is the required solution.