A satellite of \( 10^3 \, \text{kg} \) mass is revolving in a circular orbit of radius \( 2R \). If \[ \frac{10^4 R}{6} \, \text{J} \] energy is supplied to the satellite, it would revolve in a new circular orbit of radius: \[ \text{(use } g = 10 \, \text{m/s}^2, \, R = \text{radius of earth)}. \]
- 2.5 R
- 3 R
- 4 R
- 6 R
The Correct Option is D
Approach Solution - 1
\[ \text{Total energy} = \frac{-GMm}{2(2R)} \] If energy \( \frac{10^4R}{6} \) is added, then \[ \frac{-GMm}{4R} + \frac{10^4R}{6} = \frac{-GMm}{2r} \] where \( r \) is the new radius of revolution and \( g = \frac{GM}{R^2} \). \[ -\frac{mgR}{4} + \frac{10^4R}{6} = -\frac{mgR^2}{2r} \quad (m = 10^3\, \text{kg}) \] \[ -\frac{10^3 \times 10 \times R}{4} + \frac{10^4R}{6} = -\frac{10^3 \times 10 \times R^2}{2r} \] \[ -\frac{1}{4} + \frac{1}{6} = -\frac{R}{2r} \] \[ r = 6R \]
Approach Solution -2
The total mechanical energy of a satellite in a circular orbit is given by:
\[ E = -\frac{GMm}{2r}, \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.
When energy is supplied to the satellite, the total energy increases, which results in an increase in the radius of the orbit. Let the initial radius of the orbit be \( 2R \), and the new radius be \( r' \).
Using the conservation of energy:
\[ E_{\text{initial}} + \text{Energy supplied} = E_{\text{final}}, \]
\[ -\frac{GMm}{2(2R)} + \frac{10^4R}{6} = -\frac{GMm}{2r'}. \]
Simplifying, we find that the new radius \( r' \) is:
\[ r' = 6R. \]
Thus, the correct answer is \( r' = 6R \), and the correct answer is Option (4).
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