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Chapter 8: Problem 41

Energy required to move a body of mass \(m\) from an orbit of radius \(2 R\) to \(3 R\) is (a) \(\frac{G M m}{12 R^{2}}\) (b) \(\frac{G M m}{3 R^{2}}\) (c) \(\frac{G M m}{8 R}\) (d) \(\frac{G M m}{6 R}\)

Short Answer

Expert verified
\( \Delta E = \frac{GMm}{6R}\), so the correct answer is option (d).

Step by step solution

01

Understand the Basic Formula for Gravitational Potential Energy

Gravitational potential energy of a mass \(m\) in the gravitational field of another, larger mass \(M\) can be given by \(-\frac{GMm}{r}\) where \(G\) is the gravitational constant, \(r\) is the distance from the center of the mass \(M\) and \(m\) is the smaller mass.
02

Calculate the Initial and Final Energy in the two orbits

Using the formula above, we can calculate the initial energy (\(E1\)) and the final energy (\(E2\)) of the mass \(m\) in the two orbits. The initial orbit radius is \(2R\), so \(E1 = -\frac{GMm}{2R}\). The final orbit radius is \(3R\), so \(E2 = -\frac{GMm}{3R}\).
03

Calculate the Required Energy to Move the Mass

The energy required to move the mass from the initial to the final orbit is given by the difference in the energies, so \(\Delta E = E2 - E1\). Substituting the values we got in step 2, we have \(\Delta E = -\frac{GMm}{3R}- (-\frac{GMm}{2R}) = \frac{GMm}{6R}\).

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