A satellite of mass m is Launched from the earth's surface in a circular orbit. If M and R denote the mas and radius of the earth respectively, then the total energy of the satellite at an altitude 2R is

Options

A. `(GMm)/(6R)`
B. `(GMm)/(3R)`
C. `(GMm)/(2R)`
D. `(GMm)/(6R)`

Explanation

To find the total energy of a satellite at an altitude of 2R above the Earth's surface, we can follow these steps:

### Step 1: Determine the distance from the center of the Earth
The altitude of the satellite is given as 2R. Therefore, the distance from the center of the Earth to the satellite (denoted as \(d\)) is:
\[ d = R + 2R = 3R \]

### Step 2: Calculate the gravitational potential energy (U)
The gravitational potential energy (U) of the satellite at a distance \(d\) from the center of the Earth is given by the formula:
\[ U = -\frac{GMm}{d} \]
Substituting \(d = 3R\):
\[ U = -\frac{GMm}{3R} \]

### Step 3: Calculate the orbital speed (v) of the satellite
The orbital speed of a satellite in a circular orbit is given by:
\[ v = \sqrt{\frac{GM}{d}} \]
Substituting \(d = 3R\):
\[ v = \sqrt{\frac{GM}{3R}} \]

### Step 4: Calculate the kinetic energy (K)
The kinetic energy (K) of the satellite is given by:
\[ K = \frac{1}{2} mv^2 \]
Substituting for \(v\):
\[ K = \frac{1}{2} m \left(\frac{GM}{3R}\right) = \frac{GMm}{6R} \]

### Step 5: Calculate the total energy (E)
The total energy (E) of the satellite is the sum of its kinetic energy and gravitational potential energy:
\[ E = K + U \]
Substituting the expressions for \(K\) and \(U\):
\[ E = \frac{GMm}{6R} - \frac{GMm}{3R} \]

### Step 6: Simplify the expression for total energy
To combine the terms, we need a common denominator:
\[ E = \frac{GMm}{6R} - \frac{2GMm}{6R} = \frac{GMm}{6R} - \frac{2GMm}{6R} = -\frac{GMm}{6R} \]

### Final Answer
Thus, the total energy of the satellite at an altitude of 2R is:
\[ E = -\frac{GMm}{6R} \]

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