Find the total mechanical energy of a satellite of mass \(m\) revolving in a circular orbit of radius \(a\) around the Earth (mass \(M\)).
Show Hint
A negative total energy signifies that the satellite is "bound" to the Earth. If the energy were zero or positive, the satellite would have enough energy to escape the Earth's gravitational pull.
Step 1: Understanding the Concept:
The total mechanical energy of an orbiting satellite is the algebraic sum of its kinetic energy ($K$) and potential energy ($U$). The satellite requires a centripetal force to maintain its circular orbit, which is provided by the gravitational pull of the Earth. Step 2: Key Formula or Approach:
Equating gravitational force to centripetal force gives kinetic energy:
\[ \frac{GMm}{a^2} = \frac{mv^2}{a} \implies K = \frac{1}{2}mv^2 = \frac{GMm}{2a} \]
Potential Energy ($U$) $= -\frac{GMm}{a}$.
Total Energy ($E$) $= K + U$. Step 3: Detailed Explanation:
Substitute the values of $K$ and $U$ into the total energy equation:
\[ E = \frac{GMm}{2a} + \left( -\frac{GMm}{a} \right) \]
To add them, find a common denominator:
\[ E = \frac{GMm}{2a} - \frac{2GMm}{2a} = -\frac{GMm}{2a} \]
The negative sign indicates that the satellite is bound to the Earth. Step 4: Final Answer:
The total mechanical energy is $-\frac{GMm}{2a}$.