Two satellite A and B, ratio of masses `3:1` are in circular orbits of radii `r and 4r`. Then ratio mechanical energy of A and B is

Options

A. `1:3`
B. `3:1`
C. `3:4`
D. `12:1`

Explanation

To find the ratio of the mechanical energy of satellites A and B, we can follow these steps:

### Step 1: Understand the formula for mechanical energy of a satellite
The mechanical energy (E) of a satellite in a circular orbit is given by the formula:
\[ E = -\frac{G M m}{2r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( m \) is the mass of the satellite,
- \( r \) is the radius of the orbit.

### Step 2: Write the expressions for mechanical energy of satellites A and B
For satellite A:
\[ E_A = -\frac{G M m_A}{2r_A} \]
For satellite B:
\[ E_B = -\frac{G M m_B}{2r_B} \]

### Step 3: Substitute the known values
Given:
- The mass ratio \( \frac{m_A}{m_B} = \frac{3}{1} \)
- The radius of satellite A's orbit \( r_A = r \)
- The radius of satellite B's orbit \( r_B = 4r \)

Substituting these into the energy equations:
\[ E_A = -\frac{G M m_A}{2r} \quad \text{and} \quad E_B = -\frac{G M m_B}{2(4r)} \]

### Step 4: Simplify the expressions
Now substituting the mass ratio:
\[ E_A = -\frac{G M (3m_B)}{2r} \quad \text{and} \quad E_B = -\frac{G M m_B}{2(4r)} \]

### Step 5: Calculate the ratio of mechanical energies
Now we can find the ratio \( \frac{E_A}{E_B} \):
\[ \frac{E_A}{E_B} = \frac{-\frac{G M (3m_B)}{2r}}{-\frac{G M m_B}{8r}} = \frac{3m_B}{\frac{m_B}{4}} = 3 \times 4 = 12 \]

### Step 6: State the final ratio
Thus, the ratio of the mechanical energy of satellite A to satellite B is:
\[ \frac{E_A}{E_B} = 12:1 \]