A satellite A of mass m is at a distance of r from the centre of the earth. Another satellite B of mass 2m is at distance of 2r from the earth's centre. Their time periode are in the ratio of

Options

A. `1:2`
B. `1:16`
C. `1:32`
D. `1:2sqrt(2)`

Explanation

To find the ratio of the time periods of two satellites A and B, we will use the formula for the time period of a satellite in orbit around the Earth.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Satellite A has mass \( m \) and is at a distance \( r \) from the center of the Earth.
- Satellite B has mass \( 2m \) and is at a distance \( 2r \) from the center of the Earth.

2. **Formula for Time Period of a Satellite:**
The time period \( T \) of a satellite in a circular orbit is given by:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

3. **Calculate Time Period for Satellite A:**
For satellite A, the distance from the center of the Earth is \( r \):
\[ T_A = 2\pi \sqrt{\frac{r^3}{GM}} \]

4. **Calculate Time Period for Satellite B:**
For satellite B, the distance from the center of the Earth is \( 2r \):
\[ T_B = 2\pi \sqrt{\frac{(2r)^3}{GM}} = 2\pi \sqrt{\frac{8r^3}{GM}} = 2\pi \cdot 2\sqrt{\frac{r^3}{GM}} = 4\pi \sqrt{\frac{r^3}{GM}} \]

5. **Find the Ratio of the Time Periods:**
Now, we will find the ratio \( \frac{T_A}{T_B} \):
\[ \frac{T_A}{T_B} = \frac{2\pi \sqrt{\frac{r^3}{GM}}}{4\pi \sqrt{\frac{r^3}{GM}}} \]
The \( 2\pi \) and \( \sqrt{\frac{r^3}{GM}} \) terms cancel out:
\[ \frac{T_A}{T_B} = \frac{1}{2} \]

6. **Final Result:**
The ratio of the time periods of satellites A and B is:
\[ T_A : T_B = 1 : 2 \]