The time period of an artificial satellite in a circular orbit of radius `R` is `2` days and its orbital velocity is `v_(0)`. If time period of another satellite in a circular orbit is `16` days then

Options

A. its radius of orbit is 4R and orbital velocity is `v_(0)`
B. its radius of orbit is 4R and orbital velocity is `(v_(0))/(2)`
C. its radius of orbit is 2R and orbital velocity is `v_(0)`
D. its radius of orbit is 2R and orbital velocity is `(v_(0))/(2)`

Answer

B

Explanation

(b) According to Kepler's third law
`T^(2) prop R^(3)`
` :. (T_(1))/(T_(2))=((R_(1))/(R_(2)))^(3//2) or R_(2)=(R_(1))((T_(2))/(T_(1)))^(2//3)=(R_(1))((16)/(2))^(2//3)`
`= 4R_(1)=4R" "`(given `R_(1)=R`)…… (i)
Orbital velocity, `v_(o)=sqrt((GM)/(r ))`
` :. (v_(o2))/(v_(o1))=sqrt((R_(1))/(R_(2)))=sqrt((R_(1))/(4R_(1)))=(1)/(2)" "` (using (i))
`or v_(o2)=(1)/(2)v_(o1)=(1)/(2) v_(o)`