Question

An artificial satellite moving in a circular orbit around the earth has a total (kinetic + potential) energy $E₀$, its potential energy is

• A. $-E₀$
• B. $1.5 E₀$
• C. $2 E₀$
• D. $E₀$

Hint:

For circular orbit, $PE = -2KE$ and $E = -KE = PE/2$.

Answer 1

The Correct Option is C

To solve the problem, we need to understand the relationship between the kinetic energy, potential energy, and the total energy of an artificial satellite orbiting the Earth. In a circular orbit:

1. \(E_{\text{total}} = E_{\text{kinetic}} + E_{\text{potential}}\)
2. For a gravitational system such as a satellite in orbit, the total energy \(E_0\) is given by the equation:

• \(E_{\text{total}} = -\frac{GMm}{2r}\)

1. The kinetic energy (\(E_{\text{kinetic}}\)) of the satellite is:

• \(E_{\text{kinetic}} = \frac{GMm}{2r}\)

1. Therefore, the potential energy (\(E_{\text{potential}}\)) of the satellite is:

• \(E_{\text{potential}} = -\frac{GMm}{r}\)

1. The total mechanical energy \((E_0)\) of the satellite is:

• \(E_{\text{total}} = E_{\text{kinetic}} + E_{\text{potential}} = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r}\)

1. So, \(E_{\text{total}} = E_0\) is equivalent to \(-\frac{GMm}{2r}\).
2. From the above equations, we know:

• \(E_{\text{potential}} = 2 \cdot E_{\text{total}} = 2 \cdot E_0\)

1. Thus, the potential energy of the satellite is \(2 E_0\).

Conclusion: The answer is option \(2 E_0\), matching with the correct answer provided.