Question

To find some of the parameters characterizing an object moving in a circular orbit.The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit: a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass M.For all parts of this problem, where appropriate, use G for the universal gravitational constant.The potential energy U of an object of mass m that is separated by a distance R from an object of mass M is given byU=−GMmR.What is the kinetic energy K of the satellite?Find an expression for the square of the orbital period.Express your answer in terms of G, M, R, and π.

Answer 1

Answer:

$K = \frac{GMm}{2r}$

$T^2 = 4\pi^2(\frac{r^3}{GM})$

Explanation:

As we know that for a satellite the force of gravitation is equal to the centripetal force

so we will have

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

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

so we know that kinetic energy is given as

$K = \frac{1}{2}mv^2$

so we have

$K = \frac{GMm}{2r}$

now for time period we know

$T = \frac{2\pi r}{v}$

from above expression of kinetic energy we have

$v = \sqrt{\frac{GM}{r}}$

so we have

$T = \frac{2\pi r}{\sqrt{\frac{GM}{r}}}$

so square of time period is given as

$T^2 = 4\pi^2(\frac{r^3}{GM})$