= Solution
Use the figure's upward-positive vertical coordinate. The support is at $(x,0)$ and the bob at $(x+l\sin\phi,-l\cos\phi)$. Thus
$$
v_2^2=(\dot x+l\dot\phi\cos\phi)^2+(l\dot\phi\sin\phi)^2
=\dot x^2+2l\dot x\dot\phi\cos\phi+l^2\dot\phi^2.
$$
The potential is $V=-m_2gl\cos\phi$, up to an irrelevant constant. Combining both <kinetic energies> gives
$$
\boxed{L=\frac12(m_1+m_2)\dot x^2+
\frac12m_2l^2\dot\phi^2+m_2l\dot x\dot\phi\cos\phi+m_2gl\cos\phi.}
$$
The cross term expresses the moving support's contribution to the bob <velocity>; it would be lost by treating the pivot as fixed.
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