Find the orbiting velocity of a satellite revolving at a height 'h' from the surface of earth. Also calculate time period of the satellite & show that it follows Kepler's 3rd law. [Mass of earth =M, Radius of earth=R.
Answers
Orbital Speed Equation
Consider a satellite with mass Msat orbiting a central body with a mass of mass MCentral. The central body could be a planet, the sun or some other large mass capable of causing sufficient acceleration on a less massive nearby object. If the satellite moves in circular motion, then the net centripetal force acting upon this orbiting satellite is given by the relationship
Fnet = ( Msat • v2 ) / R
This net centripetal force is the result of the gravitational force that attracts the satellite towards the central body and can be represented as
Fgrav = ( G • Msat • MCentral ) / R2
Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force can be set equal to each other. Thus,
(Msat • v2) / R = (G • Msat • MCentral ) / R2
Observe that the mass of the satellite is present on both sides of the equation; thus it can be canceled by dividing through by Msat. Then both sides of the equation can be multiplied by R, leaving the following equation.
v2 = (G • MCentral ) / R
Taking the square root of each side, leaves the following equation for the velocity of a satellite moving about a central body in circular motion
where G is 6.673 x 10-11 N•m2/kg2, Mcentral is the mass of the central body about which the satellite orbits, and R is the radius of orbit for the satellite.