Question

Problem 3 The planet shown has a mass of M, and the satellite is in a circular orbit of radius r a) In terms of M, r, and the universal gravitational constant G, what is the period T of the satellite? Derive the formula b) By what factor would the period change if the mass of the planet doubled? c) By what factor would the period change if the radius of the orbit doubled?

Answer 1

Given

mass of the planet is M ,

radius of the orbit is r

the centripetal force = gravitational force

mv^2/r = G*m*M/r^2

v = sqrt(G*M/r)

that is the orbitla speed of the satellite is  

v = sqrt(G*M/r)

and the velocity v = 2pi*r/T

a)

T = 2pi*r/v

T = 2pi*r/sqrt(G*M/r)

squaring on both sides

T^2 = 4pi^2*r^3/G*M

b) if the mass of the planet is doubled then

T2^2/T1^2 = M1/2M1

T2^2 = T1^2/2

T2 = sqrt(T1^2/2)

T2 = T1/sqrt(2)

T2= 0.707*T1

if the mass of the planet doubles then the time period becomes the 0.707 of T1

c) if the radiusof the orbit doubles then

r2 = 2r1

T^2 = 4pi^2*r^3/G*M

T2^2/T1^2 = r2^3/r1^3

T2^2 = ((2r1)^3/r1^3 ) T1^2

T2 = sqrt(8)* T1

T2 = 2.83 *T1

if the radius of the orbit doubles then the time period becomes the 2.83 times the intial time period.