Question

An astronaut orbiting in a spaceship round the earth has a centripetal acceleration of $2.45{\text{m/s}}^2$. The height of the spaceship from earth's surface is
$(R=$ radius of earth $)$

• A. $3R$
• B. $2R$
• C. $R$
• D. $R/2$

Answer 1

The correct option is C $R$

For an object in circular orbit, centripetal acceleration is given by

$a_c=\frac{v^2}{r}$

And orbital velocity will be

$v=\sqrt{\frac{GM}{r}}$

$\therefore a_c=\frac{v^2}{r}=\frac{GM}{r^2}$

for $h$ height above the earth surface, $r=R+h$

$\Rightarrow a_c=\frac{GM}{(R+h{)}^2}=\frac{GM}{R^2}\times \frac{1}{(1+h/R{)}^2}$

$\Rightarrow a_c=\frac{g}{(1+h/R{)}^2}(\because g=GM/R^2)$

$\Rightarrow (1+h/R{)}^2=\frac{g}{a_c}=\frac{9.8}{2.45}=4$

$\Rightarrow h=R$

Hence, option $\left(c\right)$ is correct.

Key Concept - For a satellite in circular orbit, centripetal acceleration $a_c=\frac{v^2}{r}$