Assume that the earth moves around the sun in circular orbit of radius R and there exists a planet which also moves around the sun in a circular orbit with an radius of the orbit of the planet is

Options

A. `2^(-2//3)R`
B. `2^(2//3)R`
C. `2^(-1//3)R`
D. `(R )/(sqrt(2))`

Answer

A

Explanation

From Kepler.s third law or law of law of periods, `T^(2) prop R^(3)` Now, let us assume that radius of earth is `R_(e )` and that of other planet be `R_(rho)`. Again, time period of earth.s revolution be `T_(e )` and that of other planet.s revolution velocity of earth and planet respectively and `R_(e )=R`.
As we know, `T_(e )=(2pi)/(omega_(e ))` and `T_(p)=(2pi)/(omega_(p))`
`:. (T_(e ))/(T_(p))=(omega_(p))/(omega_(e ))" "`...(i)
Again, from Kepler.s law, we get
`(T_(e )^(2))/(T_(p)^(2))=(R_(e )^(3))/(R_(p)^(3)) rArr ((T_(e ))/(T_(p)))^(2)=((R_(e ))/(R_(p)))^(3)`
`rArr (omega_(rho))/(omega_(e ))=((R_(e ))/(R_(rho)))^(3//2)" "`[from eq. (i)]
`rArr (R )/(R_(rho))=((2omega_(e ))/(omega_(e )))^(2//3)=2^(2//3)" "[.: R_(e )=R]`
`rArr R_(rho)=2^(-2//3)R`