The minimum and maximum distances of a satellite from the center of the earth are `2R` and `4R` respectively, where `R` is the radius of earth and `M` is the mass of the earth . Find
(a) its minimum and maximum speeds,
(b) radius of curvature at the point of minimum distance.

Explanation

`(a)` Applying conservation of angular momentum
`mv_(1)(2R)=mv_(2)(4R)`
`v_(1)=2v_(2)`………`(1)`
From C.O.E.
`(1)/(2)mv_(1)^(2)-(GMm)/(2R)=(1)/(2)mv_(2)^(2)-(GMm)/(4R)`…………..`(2)`
Solving Eqs. `(1)` and `(2)`,
`v_(2)=sqrt((GM)/(6R))`, `v_(1)=sqrt((2GM)/(3R))`
`(b)` If `r` is the radius of curvature at point `B`
`(mv_(1)^(2))/(r )=(GMm)/((4R)^(2))`
`r=(16v_(2)^(2)R^(2))/(GM)=(8R)/(3)` (putting value of `v_(2)`)