Pendulum In Accelerated Frame

Inside an accelerating lift the pendulum responds to an effective gravity $g_{eff}$ that combines true gravity with the pseudo-force from the frame's acceleration. For a lift accelerating downward with acceleration $a$, the effective gravity is reduced to $g_{eff}=g-a$, since the pseudo-force points upward relative to the bob. With $a=g/2$, we get $g_{eff}=g-g/2=g/2$. The period is $T=2\pi \sqrt{L/g_{eff}}$, so reducing the effective gravity to half its value increases the period by $\sqrt{g/(g/2)}=\sqrt{2}$, since the period varies inversely with the square root of the effective gravity acting on the bob. The option claiming a decrease by $\sqrt{2}$ reverses the direction of the effect, which would correspond to upward acceleration that strengthens effective gravity. The option of no change ignores the pseudo-force entirely. The option of doubling would require $g_{eff}=g/4$, meaning downward acceleration of $3g/4$, not $g/2$. This applies the NCERT principle of effective gravity in non-inertial frames. As a plausibility check, weaker effective gravity always slows a pendulum, so the period must lengthen, and the $\sqrt{2}$ factor follows the inverse-square-root dependence on gravity.