Source: Principles of Physics
Problem
A pendulum is attached to a board which can fall freely without friction down guide ropes. Before the board is released the pendulum is deflected from the position of equilibrium (Fig. 72).
No, the pendulum will not swing.
In the reference frame of the freely falling board, the board's acceleration is $\vec{a} = \vec{g}$. An object of mass $m$ in this frame experiences an inertial pseudo-force $\vec{F}_{pseudo} = -m\vec{a} = -m\vec{g}$. The net force on the pendulum bob in this frame is the vector sum of real forces (gravity $m\vec{g}$ and tension $\vec{T}$) and the pseudo-force. The effective gravitational force on the bob is $\vec{F}_{g,eff} = m\vec{g} + \vec{F}_{pseudo} = m\vec{g} - m\vec{g} = 0$. The bob is effectively weightless. The tension in the string becomes zero, and there is no restoring force to cause oscillations. The pendulum will maintain its initial deflected position relative to the falling board.