A communication satellite of earth which takes 24 hrs. to complete one circular orbit eventually has to be replaced by another satellite of double mass. It the new satellites also has an orbital time period of 24 hrs, then what is the ratio of the radius of the new orbit to the original orbit ?

Options

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

Answer

To solve the problem, we need to determine the ratio of the radius of the new satellite's orbit to the original satellite's orbit, given that both satellites have the same orbital period of 24 hours.

### Step-by-Step Solution:

1. **Understand the relationship between orbital period and radius**:
According to Kepler's third law of planetary motion, the square of the orbital period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit. This can be expressed mathematically as:
\[ T^2 \propto r^3 \]
or
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

2. **Assign values to the periods**:
Given that both the original satellite and the new satellite have the same orbital period:
\[ T_1 = T_2 = 24 \text{ hours} \]

3. **Set up the equation**:
Since \(T_1 = T_2\), we can write:
\[ \frac{T_1^2}{T_2^2} = \frac{24^2}{24^2} = 1 \]
Therefore, we have:
\[ \frac{r_1^3}{r_2^3} = 1 \]

4. **Simplify the equation**:
From the equation \(\frac{r_1^3}{r_2^3} = 1\), we can deduce that:
\[ r_1^3 = r_2^3 \]
Taking the cube root of both sides gives:
\[ r_1 = r_2 \]

5. **Find the ratio of the radii**:
The ratio of the radius of the new orbit to the original orbit is:
\[ \frac{r_2}{r_1} = 1 \]
Thus, the ratio of the radius of the new orbit to the original orbit is:
\[ \text{Ratio} = 1:1 \]

### Final Answer:
The ratio of the radius of the new orbit to the original orbit is \(1:1\).