Question

A satellite that goes around the earth once every 24 hours is called a geosynchronous satellite. If a geosynchronous satellite is in an equatorial orbit, its position appears stationary with respect to a ground station, and it is known as a geostationary satellite

Find the radius R of the orbit of a geosynchronous satellite that circles the earth. (Note that R is measured from the center of the earth, not the surface.) You may use the following constants:
The universal gravitational constant G is
`6.67 * 10^(-11)\;{\rm N \; m^2 / kg^2}`.
The mass of the earth is
`5.98 * 10^(24)\;{\rm kg}`.
The radius of the earth is
`6.38 * 10^(6)\;{\rm m}`.

Answer 1

35870474.30504 m

Step-by-step explanation:

Given that,

r = Distance from the surface

T = Time period = 24 h

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

m = Mass of the Earth = 5.98 × 10²⁴ kg

Radius of Earth = 6.38 × 10⁶ m

The gravitational force will balance the centripetal force

`(GMm)/(R^2)=m(v^2)/(R)\\\Rightarrow v=\sqrt{(GM)/(R)}T=(2\pi r)/(v)\\\Rightarrow T=\frac{2\pi r}{\sqrt{(GM)/(r)}}`

From Kepler's law we have relation

`T^2=(4\pi^2r^3)/(GM)\\\Rightarrow r^3=(T^2GM)/(4\pi^2)\\\Rightarrow r=\left(((24* 3600)^2* 6.67* 10^(-11)* 5.98* 10^(24))/(4\pi^2)\right)^{(1)/(3)}\\\Rightarrow r=42250474.30504\ m`

Distance from the center of the Earth would be

`42250474.30504-6.38* 10^6\\`

= 35870474.30504 m