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

Options

A. (a)`1/3`
B. (b)3
C. (c)`3/4`
D. (d)12

Answer

To solve the problem of finding the ratio of total mechanical energy of satellites A and B, we can follow these steps:

### Step 1: Understand the formula for total mechanical energy of a satellite
The total 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: Define the variables for satellites A and B
Let:
- Mass of satellite A, \( m_1 = 3m \) (since the ratio of masses is 3:1),
- Mass of satellite B, \( m_2 = m \),
- Radius of orbit for satellite A, \( r_1 = r \),
- Radius of orbit for satellite B, \( r_2 = 4r \).

### Step 3: Calculate the total mechanical energy for each satellite
Using the formula for total mechanical energy:

For satellite A:
\[ E_A = -\frac{G M m_1}{2r_1} = -\frac{G M (3m)}{2r} = -\frac{3G M m}{2r} \]

For satellite B:
\[ E_B = -\frac{G M m_2}{2r_2} = -\frac{G M m}{2(4r)} = -\frac{G M m}{8r} \]

### Step 4: Find the ratio of total mechanical energy of A to B
Now, we need to find the ratio \( \frac{E_A}{E_B} \):
\[ \frac{E_A}{E_B} = \frac{-\frac{3G M m}{2r}}{-\frac{G M m}{8r}} = \frac{3G M m}{2r} \cdot \frac{8r}{G M m} \]

### Step 5: Simplify the ratio
Cancelling out the common terms \( G, M, m, \) and \( r \):
\[ \frac{E_A}{E_B} = \frac{3 \cdot 8}{2} = \frac{24}{2} = 12 \]

### Conclusion
The ratio of total mechanical energy of satellite A to satellite B is:
\[ \frac{E_A}{E_B} = 12:1 \]

### Final Answer
Thus, the ratio of total mechanical energy of A to B is \( 12:1 \).

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