Two satellites `A` and `B` of masses `m_(1)` and `m_(2)(m_(1)=2m_(2))` are moving in circular orbits of radii `r_(1)` and `r_(2)(r_(1)=4r_(2))`, respectively, around the earth. If their periods are `T_(A)` and `T_(B)`, then the ratio `T_(A)//T_(B)` is

Options

A. `4`
B. `16`
C. `2`
D. `8`

Explanation

To solve the problem, we need to find the ratio of the periods \( T_A \) and \( T_B \) of two satellites \( A \) and \( B \) orbiting the Earth. We have the following information:

1. Mass of satellite \( A \): \( m_1 = 2m_2 \)
2. Radius of orbit of satellite \( A \): \( r_1 = 4r_2 \)

### Step 1: Write the formula for the time period of a satellite

The time period \( T \) of a satellite in a circular orbit is given by the formula:

\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]

where \( r \) is the radius of the orbit, \( G \) is the gravitational constant, and \( M \) is the mass of the Earth.

### Step 2: Write the expressions for \( T_A \) and \( T_B \)

Using the formula, we can express the time periods for satellites \( A \) and \( B \):

\[ T_A = 2\pi \sqrt{\frac{r_1^3}{GM}} \]
\[ T_B = 2\pi \sqrt{\frac{r_2^3}{GM}} \]

### Step 3: Find the ratio \( \frac{T_A}{T_B} \)

Now, we can find the ratio of the time periods:

\[ \frac{T_A}{T_B} = \frac{2\pi \sqrt{\frac{r_1^3}{GM}}}{2\pi \sqrt{\frac{r_2^3}{GM}}} = \frac{\sqrt{r_1^3}}{\sqrt{r_2^3}} = \sqrt{\frac{r_1^3}{r_2^3}} \]

This simplifies to:

\[ \frac{T_A}{T_B} = \sqrt{\left(\frac{r_1}{r_2}\right)^3} \]

### Step 4: Substitute the known values

We know that \( r_1 = 4r_2 \). Therefore, we can substitute this into our equation:

\[ \frac{T_A}{T_B} = \sqrt{\left(\frac{4r_2}{r_2}\right)^3} = \sqrt{4^3} = \sqrt{64} = 8 \]

### Final Answer

Thus, the ratio of the periods of the two satellites is:

\[ \frac{T_A}{T_B} = 8 \]