Question

An astronaut orbiting in a spaceship around the earth has a centripetal acceleration of $2.5{\text{m/s}}^2$. The height of spaceship from earth's surface is ($R=$radius of earth and take $g=10{\text{m/s}}^2$).

• A. $3R$
• B. $2R$
• C. $R$
• D. $\frac{R}{2}$

Answer 1

The correct option is C $R$

The centripetal acceleration for an object orbiting in circular orbit at a height $h$ from the surface of the planet having radius $R$, $$a_c=\frac{g}{{(1+\frac{h}{R})}^2}$$ Substituting the values from given problem,

$2.5=\frac{10}{{(1+\frac{h}{R})}^2}$

$\Rightarrow {(1+\frac{h}{R})}^2=4$

$\Rightarrow (1+\frac{h}{R})=+2\text{or}(1+\frac{h}{R})=-2$

Positive value is taken as $h$ cannot be negative.

$\Rightarrow \frac{h}{R}=1$

$\therefore h=R$

Hence, the height of spaceship is $R$ and option (c) is correct answer.