Question

Work required to shift an artificial satellite from an orbit of radius \(r\) to an orbit of radius \(2r\) is

• A. \( \dfrac{GMm}{2r} \)
• B. \( \dfrac{GMm}{4r} \)
• C. \( \dfrac{GMm}{8r} \)
• D. Zero

Hint:

Remember: \[ E=-\frac{GMm}{2R} \] Higher orbit means higher (less negative) total energy.

Answer 1

The Correct Option is B

Step 1: Recall the satellite energy formula.
A satellite going around in a circle of radius $R$ has total energy \[ E = -\frac{GMm}{2R} \] The minus sign means it is bound to the planet.

Step 2: Link work to energy change.
To lift the satellite to a bigger orbit, the work we must do equals the change in its total energy. \[ W = E_f - E_i \]
Step 3: Energy in the starting orbit.
With radius $r$, \[ E_i = -\frac{GMm}{2r} \]
Step 4: Energy in the final orbit.
With radius $2r$, \[ E_f = -\frac{GMm}{2(2r)} = -\frac{GMm}{4r} \]
Step 5: Subtract to get the work.
\[ W = -\frac{GMm}{4r} - \left(-\frac{GMm}{2r}\right) \] \[ = -\frac{GMm}{4r} + \frac{GMm}{2r} \]
Step 6: Combine the fractions.
Using a common bottom of $4r$, \[ W = \frac{-GMm + 2GMm}{4r} = \frac{GMm}{4r} \] \[ \boxed{\dfrac{GMm}{4r}} \]