Let
#m->"mass of satellite" #
#M->"mass of Earth" #
#R->"radius of the orbit of the satellite" #
#G->"Gravitational constant" #
#T->"time period of satellite" #
#omega->"angular speed of satellite"=(2pi)/T #
The centripetal force (#F_c#) acting on the satellite revolving round the Earth along the orbit having radius R is given by
#F_c=momega^2R#
The gravitaional force #(F_g)# acting on the satellite will provide the required centripetal force.
#F_g=G(mM)/R^2#
Now #F_c=F_g#
#=>momega^2R=(GmM)/R^2#
#=>((2pi)/T)^2R=(GM)/R^2#
#=>T^2=((4pi^2)/(GM))R^3.......
(1)#
Now differentiating (1) w.r to R we get
#=>2TdT=((4pi^2)/(GM))*3R^2dR........(2)#
Dividing (2) by (1) we get
#2(dT)/T=(3dR)/R#
#=>(dT)/T=3/2((dR)/R)......(3)#
Now by the problem change in radius of the satellite is
#dR=1.02R-R=0.02R#
Inserting this value of #dR=0.02R# in equation (3) we get the percentage change in time period of second satellite w.r to the time period of first satellite.
#"% change in time period"#
#=(dT)/Txx100%=3/2((0.02R)/R)xx100%#
#=3%#
