Question

Let the period of revolution of a planet at a distance R from a star be T. prove that if it was at a distance of 2R from the star, its period of revolution will be √8T

Answer 1

Given:D=R(distance =radius of the earth)
T=Time period of revolution.
G=Universal gravitation constant.
M=Mass of the star(Sun).
R/r=Distance of planet from Sun.
1) r=R,T=T
therefore,
T1=2pi/sqrt G m...(1)
2)For r=2R,T=T2
therefore,T2=2pi/sqrtG m (2R)3/2
therefore, T2=2pi/sqrtG m multiplied by 2^3/2 multiplied by R^3/2
therefore,
T2=2pi/sqrtG m multiply by R^3/2 multiply by 2^3/2
= T2=2pi/sqrtG M R^3/2 multiply by sqrt8
T1=2pi/sqrtG mR^3/2
therefore,
T2=sqrt8 T1