Pendulum with a freely sliding support ($M_{\mathrm{eff}}(\varphi )=m_2{l}^2(m_1+m_2{\sin }^2\varphi )/(m_1+m_2)$)

For a pendulum whose support slides without horizontal force, the support coordinate is cyclic and total horizontal momentum is conserved. At zero horizontal momentum, eliminating support velocity leaves the displayed angle-dependent effective inertia. The energy is $M_{\mathrm{eff}}{\dot{\varphi }}^2/2-m_{2}gl\cos \varphi$. Fixing the momentum before evaluating motion is distinct from fixing the support position.

Zero horizontal momentum fixes the horizontal centre of mass. The bob coordinates become $X=X_c+l{m}_1\sin \varphi /(m_1+m_2)$ and $Y=-l\cos \varphi$. Eliminating the angle gives an ellipse; an ordinary oscillation traverses only its permitted arc. The support counter-moves, so the bob's path differs from the circle about a fixed pivot.