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AIIMS 2001
Exam
AIIMS
Year
2001
Subject
Physics
Chapter
Gravitation

AIIMS 2001 · Physics · Gravitation

The orbital velocity of an artificial satellite in a circular orbit...

The orbital velocity of an artificial satellite in a circular orbit above the earth's surface at a distance equal to radius of earth is vv. For a satellite orbiting at an altitude half of earth's radius, orbital velocity is

Options

  1. A32v\frac{3}{2} v
  2. B3/2v\sqrt{3 / 2} v
  3. C2/3v2 / \sqrt{3} v
  4. D23v\frac{2}{3} v.

Correct answer

C. 2/3v2 / \sqrt{3} v

Step-by-step solution

The satellite is moving at an attitude above the surface equal to the radius of the earth RR. ∴mv022R=GMm4R2⇒v0=GM2R\begin{aligned}& \therefore \quad \frac{m v_0^2}{2 R}=G \frac{M m}{4 R^2}\\ & \Rightarrow v_0=\sqrt{\frac{G M}{2 R}} \end{aligned}Now if the satellite moves at an attitude equal to R/2R / 2 then, mv0′23R/2=GMm9×(R/2)2⇒v0′2=23⋅GMR⇒v0′=23GMR=23v02=43v0=23v0\begin{aligned}& \frac{m v_0^{\prime 2}}{3 R / 2}=G \frac{M m}{9 \times(R / 2)^2}\\ & \Rightarrow \quad v_0^{\prime 2}=\frac{2}{3} \cdot \frac{G M}{R}\\ & \Rightarrow \quad v_0^{\prime}=\frac{\sqrt{2}}{\sqrt{3}} \sqrt{\frac{G M}{R}}=\frac{\sqrt{2}}{\sqrt{3}}v_0 \sqrt{2}\\ & =\frac{\sqrt{4}}{\sqrt{3}}v_0=\frac{2}{\sqrt{3}}v_0 \\ & \end{aligned}
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