Question

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 of the Earth.) Use the following values if needed in this problem: The universal gravitational constant G is 6.67×10−11Nm2/kg2. The mass of the earth is 5.98×1024kg. The radius of the earth is 6.38×106m. Express your answer numerically in meters to three significant figures.

Answer 1

Answer:

$3.59\times 10^{7}\ m$

Explanation:

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

From Kepler's law which balances centripetal force and the force of gravity we have relation

$T^2=\dfrac{4\pi^2r^3}{GM}\\\Rightarrow r=\dfrac{T^2GM}{4\pi^2}\\\Rightarrow r=\left(\dfrac{(24\times 3600)^2\times 6.67\times 10^{-11}\times 5.98\times 10^{24}}{4\pi^2}\right)^{\frac{1}{3}}\\\Rightarrow r=42250474.30504\ m$

Distance from the center of the Earth would be

$42250474.30504-6.38\times 10^6=35870474.30504\ m=3.59\times 10^{7}\ m$