Question

Gravitation -

A planet has satellite which moves in a circular orbit of radius
r = 7200 km in a period T = 5h17m. Calculate the mass of the planet.
(G = 6.67 x 10^-11 Nm^2 / kg^2.)

Answer 1

Here is the question, please help me!

Answer 2

From Newton Law of Gravity
\[F=G \frac{ Mm }{ r^2 }\]

And because the satellite is circular motion you need centripetal force to change the direction of motion.
That is given by
\[F=m \frac{ v^2 }{ r }\]

Where r is the radical distance, m is the mass of the satellite, M is the mass of the planet and G is the gravitational constant.

The force of gravity and the centripetal force are in balance, so we have
\[G \frac{ Mm }{ r^2 }=m \frac{ v^2 }{ r }\]

We don't know the velocity. But we know the period in which takes to make one complete revolution. That will be the distance of a circle which is the circumference. It will be the rate its that will give you the velocity.

\[v=\frac{ 2 \pi r }{ T }\]

Plug v into the equation above and you have and doing some algebra we have,

\[\frac{ 4 \pi^2r^3 }{ GT^2 }=M\]

At this point you just plug in all the values you are given.

Answer 3

Actually, all you need is Kepler's law of periods, which snowsurf has derived for you. You don't need to derive it though. Look through your textbook, you must find the equation:
\[\frac{ r ^{3} }{ T ^{2} } = \frac{ GM }{ 4\pi ^{2} }\]