Solution:

The energy required to move a satellite from one orbit to another can be found using the difference in total mechanical energy between the two orbits.

The total energy \( E \) of a satellite in orbit of radius \( r \) is:

\[
E = -\frac{GMm}{2r}
\]

For the initial orbit of radius \( r \), the energy is:

\[
E_1 = -\frac{GMm}{2r}
\]

For the final orbit of radius \( \frac{3r}{2} \), the energy is:

\[
E_2 = -\frac{GMm}{2 \times \frac{3r}{2}} = -\frac{GMm}{3r}
\]

The energy required to shift the satellite is the difference between the two energies:

\[
\Delta E = E_2 - E_1 = \left(-\frac{GMm}{3r}\right) - \left(-\frac{GMm}{2r}\right)
\]

\[
\Delta E = \frac{GMm}{2r} - \frac{GMm}{3r} = \frac{GMm}{6r}
\]

So, the energy required is:

\[
\Delta E = \frac{GMm}{6r}
\]