Question

GEO STATIONARY AND POLAR SATELLITES
A geo-stationary satellite is orbittingthe earth at a height 6R above the surface of the earth.where R is the radius of earth.The time period of another satellite revolving around earth at a height 2.5R from earths surface is ----------

Answer 1

$\begin{array}{l}\text{Here,}\\ \text{Radius of geostationary orbit = 6R}\\ \text{Radius of the other satellite = 2}\text{.5R}\\ \text{Let the angular velocity of the geostationary satellite and the other satellite}\\ \text{respectively be}\omega \text{and}\omega \text{'}\\ \text{radius of the orbit of the geostationary satellite is given by}\\ 6R=\sqrt[3]{\frac{GM}{{\omega }^2}}---1.\\ \text{Similarly, for the other satellite}\\ 2.5R=\sqrt[3]{\frac{GM}{{(\omega \text{'})}^2}}---2.\\ \text{Dividing eqn}\text{. 1 by eqn}\text{. 2}\\ =>\sqrt[3]{\frac{{(\omega \text{'})}^2}{{\omega }^2}}=2.4\\ =>\frac{\omega \text{'}}{\omega }=3.72\\ =>\omega \text{'}=3.72\omega \\ =>\frac{2\pi }{T\text{'}}=3.72\times \frac{2\pi }{T}\\ =>T\text{'}=\frac{T}{3.72}\\ \text{We know T for geostationary satellite = 24 hr}\\ \text{=>}T\text{'}=\frac{24}{3.72}=6.45\text{hr}\end{array}$