Question

An artificial satellite is in a circular orbit around a planet of radius r = 2.25 × 10³ km at a distance d = 380.0 km from the planet\'s surface. The period of revolution of the satellite around the planet is T = 1.15 hours. What is the average density of the planet?

Answer 1

given r= 2.25 x 10³ km

Using Orbital Period Equation

T²/R³ = (4 x П²)/ (G x M_central)

|Where,
T = period of the satellite
Mcentral = mass of the central body about which the satellite orbits

G = Gravitational constant ( which equals 6.673 x 10^-¹¹ N m²/kg² )
П = 3.14

Given that
T = 1.15 hours = 4140 seconds
R = ( Radius of planet + distance of satellite from planet's surface)
= ( 2.25 x 10³) + 380
= 2630 km
= 2630000 meters
M_central = (4 x П² x R³)/ (G x T²)
= [4 x (3.14)² x (2630000)³ ] / [ 6.673×10¯¹¹ x (4140)² ]

= 6.273 x 10²³ kg

Volume of planet(V) = 4/3 x П x R³


.'. V = 4/3 x 3.14 x (2630000)³
= 7.616 x 10¹⁹ metre³

Hence, average density of planet = Mass/volume
= M_central/ Volume
= (6.273 x 10²³ kg) / (7.616 x 10¹⁹ metre³)
= 8236.607 Kg/m³