Two satellites P and Q ratio of masses `3:1` are in circular orbits of radii `r and 8r`. Then ratio of total mechanical energy of A to B is

Options

A. `1:3`
B. `3:1`
C. `3:4`
D. `24:1`

Explanation

To solve the problem, we need to find the ratio of the total mechanical energy of two satellites P and Q, given their masses and the radii of their orbits.

### Step-by-Step Solution:

1. **Identify the given data:**
- Mass of satellite P, \( m_P = 3m \)
- Mass of satellite Q, \( m_Q = m \)
- Radius of orbit of satellite P, \( r_P = r \)
- Radius of orbit of satellite Q, \( r_Q = 8r \)

2. **Recall the formula for total mechanical energy (E) of a satellite in orbit:**
The total mechanical energy \( E \) of a satellite in a circular orbit is given by:
\[ E = -\frac{GMm}{2r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the central body (e.g., Earth), \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

3. **Calculate the total mechanical energy of satellite P:**
Using the formula:
\[ E_P = -\frac{GM m_P}{2 r_P} = -\frac{GM (3m)}{2 r} = -\frac{3GMm}{2r} \]

4. **Calculate the total mechanical energy of satellite Q:**
Using the formula:
\[ E_Q = -\frac{GM m_Q}{2 r_Q} = -\frac{GM m}{2 (8r)} = -\frac{GMm}{16r} \]

5. **Find the ratio of total mechanical energies \( \frac{E_P}{E_Q} \):**
\[ \frac{E_P}{E_Q} = \frac{-\frac{3GMm}{2r}}{-\frac{GMm}{16r}} = \frac{3GMm}{2r} \times \frac{16r}{GMm} \]
Simplifying this gives:
\[ \frac{E_P}{E_Q} = \frac{3 \times 16}{2} = \frac{48}{2} = 24 \]

6. **Conclusion:**
The ratio of the total mechanical energy of satellite P to satellite Q is:
\[ \frac{E_P}{E_Q} = 24:1 \]

### Final Answer:
The ratio of total mechanical energy of satellite P to satellite Q is \( 24:1 \).