An artificial satellite revolves around a planet for which gravitational force (F) varies with distance r from its centre as F propto r2. If v0 its orbital speed, then

Question

An artificial satellite revolves around a planet for which gravitational force (F) varies with distance r from its centre as F propto r2. If v0 its orbital speed, then (A) v0 propto r-1 / 2 (B) v0 propto r3 / 2 (C) v0 propto r-3 / 2 (D) v0 propto r.

Tardigrade Admin · Accepted Answer

Gravitational force (F) provides the necessary centripetal force to keep the satellite in orbit, ⇒ (m v02/r) propto F (m v02/r) propto r2 v0 arrow Orbital speed r arrow Radius of orbit ⇒ v0 propto r3 / 2

Q. An artificial satellite revolves around a planet for which gravitational force $(F)$ varies with distance $r$ from its centre as $F \propto r^{2}$ . If $v_{0}$ its orbital speed, then

$v_{0} \propto r^{- 1/2}$

$v_{0} \propto r^{3/2}$

$v_{0} \propto r^{- 3/2}$

$v_{0} \propto r$ .

Gravitational force $(F)$ provides the necessary centripetal force to keep the satellite in orbit,
$\Rightarrow \frac{m v _{0}^{2}}{r} \propto F$
$\frac{m v _{0}^{2}}{r} \propto r^{2}$
$v_{0} \to$ Orbital speed
$r \to$ Radius of orbit
$\Rightarrow v_{0} \propto r^{3/2}$

Q. An artificial satellite revolves around a planet for which gravitational force (F) varies with distance r from its centre as F∝r2. If v0​ its orbital speed, then

v0​∝r−1/2

v0​∝r3/2

v0​∝r−3/2

v0​∝r.