Question 434: Gravitation

Question

A geostationary satellite revolves around the Earth in a circular orbit of radius $$4 R$$. Here, $R$ is the radius of the Earth. Then, the time period of another satellite moving in a circular orbit of radius $$2 R$$ is:

Accepted Answer

Let the time period of the given
geostationary satellite
be $ T_1 $ and its orbital radius be $ 4R $.
We are asked to find the time period $ T_2 $ of another satellite that is revolving in an orbit of radius $ 2R $.
Step 1: Relation between orbital period and radius
For a satellite revolving around the Earth under the influence of gravity, the time period $ T $ is related to the orbital radius $ r $ by KeplerÃ¢ÂÂs
third law
:
$ T^2 \propto r^3 $
Thus, for two satellites,
$ \frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3} $
Step 2: Substitute given values
From the question:
$ r_1 = 4R, \quad r_2 = 2R $
Substitute in the above relation:
$ \frac{T_2^2}{T_1^2} = \left( \frac{2R}{4R} \right)^3 = \left( \frac{1}{2} \right)^3 = \frac{1}{8} $
Step 3: Simplify to find $ T_2 $
$ T_2 = T_1 \sqrt{\frac{1}{8}} = \frac{T_1}{2\sqrt{2}} $
Step 4: Select the correct option
Hence, the time period of the second satellite is
$ \boxed{T_2 = \dfrac{T_1}{2\sqrt{2}}} $
Correct Option: D

Answer

$$2 T_1$$

Answer

$$2 \sqrt{2} T_1$$

Answer

$$T_1 / 2$$

Answer

$$T_1 / 2 \sqrt{2}$$

Gravitation

A geostationary satellite revolves around the Earth in a circular orbit of radius $4 R. Here, R is the radius of the Earth. Then, the time period of another satellite moving in a circular orbit of radius 2 R$ is:

Choose the correct answer: