A satallite of mass `m`, initally at rest on the earth, is launched into a circular orbit at a height equal to the the radius of the earth. The minimum energy required is

Options

A. `sqrt3/4mgR`
B. `1/2mgR`
C. `1/4mgR`
D. `3/4mgR`

Answer

To find the minimum energy required to launch a satellite of mass \( m \) into a circular orbit at a height equal to the radius of the Earth, we can follow these steps:

### Step 1: Understand the Problem
The satellite is launched into a circular orbit at a height equal to the radius of the Earth. Therefore, the total distance from the center of the Earth to the satellite in orbit is \( 2R \), where \( R \) is the radius of the Earth.

### Step 2: Calculate the Gravitational Force
The gravitational force acting on the satellite when it is at a distance \( 2R \) from the center of the Earth is given by:
\[ F_g = \frac{GMm}{(2R)^2} = \frac{GMm}{4R^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

### Step 3: Calculate the Centripetal Force
For the satellite to maintain a circular orbit, the gravitational force must equal the centripetal force required to keep it in orbit:
\[ F_c = \frac{mv^2}{2R} \]
Setting \( F_g = F_c \):
\[ \frac{GMm}{4R^2} = \frac{mv^2}{2R} \]

### Step 4: Solve for the Orbital Speed \( v \)
Canceling \( m \) from both sides and rearranging gives:
\[ v^2 = \frac{GM}{2R} \]
Thus, the speed \( v \) is:
\[ v = \sqrt{\frac{GM}{2R}} \]

### Step 5: Calculate the Kinetic Energy in Orbit
The kinetic energy \( K \) of the satellite in orbit is given by:
\[ K = \frac{1}{2} mv^2 = \frac{1}{2} m \left(\frac{GM}{2R}\right) = \frac{GMm}{4R} \]

### Step 6: Calculate the Potential Energy in Orbit
The gravitational potential energy \( U \) of the satellite at a distance \( 2R \) from the center of the Earth is:
\[ U = -\frac{GMm}{2R} \]

### Step 7: Calculate the Total Energy in Orbit
The total mechanical energy \( E \) in orbit is the sum of kinetic and potential energy:
\[ E = K + U = \frac{GMm}{4R} - \frac{GMm}{2R} \]
Combining these gives:
\[ E = \frac{GMm}{4R} - \frac{2GMm}{4R} = -\frac{GMm}{4R} \]

### Step 8: Calculate the Initial Energy
The initial energy when the satellite is at rest on the surface of the Earth (at distance \( R \)) is:
\[ E_{initial} = -\frac{GMm}{R} \]

### Step 9: Calculate the Minimum Energy Required
The minimum energy required to launch the satellite into orbit is the difference between the total energy in orbit and the initial energy:
\[ \Delta E = E - E_{initial} = -\frac{GMm}{4R} + \frac{GMm}{R} \]
Calculating this gives:
\[ \Delta E = \left(-\frac{GMm}{4R} + \frac{4GMm}{4R}\right) = \frac{3GMm}{4R} \]

### Step 10: Substitute \( g \)
We know that \( g = \frac{GM}{R^2} \), thus:
\[ \Delta E = \frac{3}{4} mgR \]

### Conclusion
The minimum energy required to launch the satellite into orbit is:
\[ \Delta E = \frac{3}{4} mgR \]

### Final Answer
The minimum energy required is \( \frac{3}{4} mgR \).

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