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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.

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(a) Applying conservation of angular momentum
`m upsilon_(1) (2R) = m upsilon_(2) (4R)`
`upsilon_(1) = 2 upsilon_(2)` ..(i)
From conservation of energy
`(1)/(2) m upsilon_(1)^(2) - (GMm)/(2R) = (1)/(2) m upsilon_(2)^(2) - (GMm)/(4R)` ..(ii)
Solving Eqs. (i) and (ii), we get
`upsilon_(2) = sqrt((GM)/(6R)), upsilon_(1) = sqrt((2Gm)/(3R))`
(b) If `r` is the radius of curvature at point `A`
`(m upsilon_(1)^(2))/(r) = (GMm)/((2R)^(2))`
`r = (4 upsilon_(1)^(2)R^(2))/(GM) = (8R)/(3)` (putting value of `upsilon_(1)`)
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