Question

The orbital speed of a body revolving very close to the surface of a planet, of radius $R$ and mean density $\rho$, is

• A. $\sqrt{\frac{\pi \rho G}{3}}$
  
• B. $\frac{R}{2}\sqrt{\frac{\pi \rho G}{3}}$
  
• C. $2R\sqrt{\frac{\pi \rho G}{3}}$
  
• D. $\frac{R}{4}\sqrt{\frac{\pi \rho G}{3}}$
  

Answer 1

The correct option is C $2R\sqrt{\frac{\pi \rho G}{3}}$


Orbital speed of a body is given by $V_o=\sqrt{\frac{GM}{r}}$

where $r=R+h$

$h\to$ height of the body from the surface of planet

since body is revolving very close to the surfce of the planet
$h<<R$

$\therefore V_o=\sqrt{\frac{GM}{R}}$

Mass of the planet, $M=\text{volume}\times \text{density}$

$M=\frac{4}{3}\pi R^3\times \rho$

$\therefore V_o=\sqrt{\frac{G4\pi R^3\rho }{3R}}$

$V_o=2R\sqrt{\frac{\pi \rho G}{3}}$

Hence, option $\left(C\right)$ is the correct answer.

Why this Question? Note: Orbital speed of a body is given by $V_o=\sqrt{\frac{GM}{r}}$ where $r=R+h$ Caution: $r$ is always measured from the centre of the planet.