Question

A satellite can be in a geostationary orbit around the earth at a distance r from the center. If the angular velocity of the earth about its axis doubles, a satellite can now be in a geostationary orbit around the earth if its distance from the center is,

Answer 1

Angular velocity

1. The angular velocity is the time rate of change of angular displacement of a moving body.
2. The angular velocity of the satellite is defined by the form, $\omega =\sqrt{\left(\frac{GM}{r^3}\right)}$, where, r is the distance of the satellite from the earth, G is the universal gravitational constant, and M is the mass of the satellite.
3. The angular velocity of a geostationary satellite is the same as the earth's angular velocity.

Step 1: Given data

1. The distance of the satellite from the earth's center is $r$.

Step 2: Finding the distance

Let the distance of the satellite after increasing angular velocity is R.

Initially, the angular velocity of the satellite is

${\omega }_i=\sqrt{\left(\frac{GM}{r^3}\right)}.............\left(1\right)$

Finally, the angular velocity is doubled. Now angular velocity is,

$2{\omega }_i=\sqrt{\left(\frac{GM}{R^3}\right)}.............\left(2\right)$

Comparing, equations 1 and 2 we get,

$\sqrt{\left(\frac{GM}{r^3}\right)}=$ $\frac{1}{2}\sqrt{\left(\frac{GM}{R^3}\right)}$

$$\begin{gathered} or\frac{1}{r^3}=\frac{1}{4}\frac{1}{R^3} \\ or4R^3=r^3 \\ or{R}^3=\frac{r^3}{4} \\ orR=\frac{r}{4^{{\displaystyle \frac{1}{3}}}} \end{gathered}$$

Therefore, the distance of the satellite after increasing angular velocity is $\frac{r}{4^{{\displaystyle \frac{1}{3}}}}$.