Step 1: Understanding the Concept:
For a satellite orbiting the Earth in a circular orbit, the total mechanical energy $E$ is the sum of its kinetic and potential energies. Since the gravitational force provides the necessary centripetal force, the total mechanical energy depends solely on the radius of the orbit.
Step 2: Key Formula or Approach:
The total mechanical energy $E$ of a satellite of mass $m$ orbiting a planet of mass $M$ at a radius $r$ is given by the formula:
\[ E = -\frac{GMm}{2r} \]
where $G$ is the universal gravitational constant.
Step 3: Detailed Explanation:
Let the initial radius be $r$. The initial total mechanical energy is:
\[ E = -\frac{GMm}{2r} \]
When the satellite dissipates energy due to resistive forces, its orbital radius decreases. The new radius is given as $r' = \frac{r}{2}$.
The new total mechanical energy $E'$ becomes:
\[ E' = -\frac{GMm}{2r'} = -\frac{GMm}{2 \left( \frac{r}{2} \right)} \]
\[ E' = -\frac{2GMm}{2r} = 2 \left( -\frac{GMm}{2r} \right) \]
Substituting the initial energy $E$:
\[ E' = 2E \]
Since $E$ is negative, $2E$ represents a more negative value, signifying a decrease in the total energy (energy dissipated).
Step 4: Final Answer:
The new total mechanical energy of the satellite is $2E$.