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7.Gravitation
hard
A satellite of mass $m,$ initially at rest on the earth, is launched into a circular orbit at a height equal to the radius of the earth. The minimum energy required is
A
$\frac{{\sqrt 3 }}{4}mgR$
B
$\frac{1}{2}mgR$
C
$\frac{1}{4}mgR$
D
$\frac{3}{4}mgR$
Solution
We know
$V_{0}=\sqrt{\frac{G M}{r}} \& g=\frac{G M}{R^{2}}$
From energy conservation
$U_{i}+K_{i}=U_{f}+K_{f}$
$-\frac{G M m}{R}+K_{f}=-\frac{G M m}{2 R}+\frac{1}{2} m v_{0}^{1}$
$K_{i}=\frac{G M m}{2 R}+\frac{1}{2} m(\sqrt{\frac{G M}{2 R}})^{2} \Rightarrow K_{i}=\frac{3 G M m}{4 R} \rightarrow K_{i}=\frac{3}{4} m g R$
Standard 11
Physics
