A skylab of mass `m` kg is first launched from the surface of the earth in a circular orbit of radius `2R` (from the centre of the earth) and then it is shifted from this circular orbit to another circular orbit of radius `3R`. The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to the second orbit are

Options

A. `3/4mgR,(mgR)/6`
B. `3/4mgR,(mgR)/12`
C. `mgR,mgR`
D. `2mgR,mgR`

Explanation

To solve the problem, we need to calculate the minimum energy required to launch the Skylab from the surface of the Earth to a circular orbit of radius \(2R\) and then the energy required to shift it from that orbit to another circular orbit of radius \(3R\).

### Step 1: Calculate the Total Energy in the Orbit of Radius \(2R\)

The total mechanical energy \(E\) of an object in a circular orbit is given by:
\[ E = -\frac{G M m}{2r} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, \(m\) is the mass of the Skylab, and \(r\) is the radius of the orbit.

For the first orbit at radius \(2R\):
\[ E_{2R} = -\frac{G M m}{2 \cdot 2R} = -\frac{G M m}{4R} \]

### Step 2: Calculate the Total Energy on the Surface of the Earth

The potential energy \(U\) on the surface of the Earth is given by:
\[ U = -\frac{G M m}{R} \]
Thus, the total energy at the surface (which is only potential energy) is:
\[ E_{surface} = -\frac{G M m}{R} \]

### Step 3: Calculate the Minimum Energy Required to Launch to \(2R\)

The minimum energy required to launch the Skylab to the first orbit is the difference in total energy between the orbit and the surface:
\[ E_{launch} = E_{2R} - E_{surface} \]
Substituting the values we calculated:
\[ E_{launch} = \left(-\frac{G M m}{4R}\right) - \left(-\frac{G M m}{R}\right) \]
\[ E_{launch} = -\frac{G M m}{4R} + \frac{G M m}{R} \]
\[ E_{launch} = \frac{G M m}{R} - \frac{G M m}{4R} = \frac{4G M m - G M m}{4R} = \frac{3G M m}{4R} \]

### Step 4: Calculate the Total Energy in the Orbit of Radius \(3R\)

For the second orbit at radius \(3R\):
\[ E_{3R} = -\frac{G M m}{2 \cdot 3R} = -\frac{G M m}{6R} \]

### Step 5: Calculate the Minimum Energy Required to Shift from \(2R\) to \(3R\)

The energy required to shift from the first orbit to the second orbit is:
\[ E_{shift} = E_{3R} - E_{2R} \]
Substituting the values we calculated:
\[ E_{shift} = \left(-\frac{G M m}{6R}\right) - \left(-\frac{G M m}{4R}\right) \]
\[ E_{shift} = -\frac{G M m}{6R} + \frac{G M m}{4R} \]
Finding a common denominator (which is \(12R\)):
\[ E_{shift} = -\frac{2G M m}{12R} + \frac{3G M m}{12R} = \frac{G M m}{12R} \]

### Final Results

1. The minimum energy required to launch the Skylab to the first orbit (\(2R\)) is:
\[ E_{launch} = \frac{3G M m}{4R} \]

2. The minimum energy required to shift the Skylab from the first orbit (\(2R\)) to the second orbit (\(3R\)) is:
\[ E_{shift} = \frac{G M m}{12R} \]