TYPES OF OSCILLATIONS

An important application of the simple-pendulum formula $T=2\pi\sqrt{l/g}$ is what happens when the point of suspension itself accelerates, since the pendulum then responds not to the true $g$ but to an effective acceleration due to gravity, $g_{\text{eff}}$. (i) If the support (e.g. a lift) accelerates upward with acceleration $a$, $g_{\text{eff}}=g+a$, so $T=2\pi\sqrt{l/(g+a)}$ -- the period decreases, since $T$ is inversely related to $g_{\text{eff}}$. (ii) If it accelerates downward with acceleration $a$ (with $a<g$), $g_{\text{eff}}=g-a$, so $T=2\pi\sqrt{l/(g-a)}$ -- the period increases. (iii) If the lift is falling with $a>g$, $g_{\text{eff}}=a-g$ and $T=2\pi\sqrt{l/(a-g)}$, but now the pendulum flips upside down and oscillates about its highest point instead of its lowest. (iv) If the lift falls freely with $a=g$ exactly, $g_{\text{eff}}=0$, so $T\to\infty$ -- the pendulum simply stops oscillating, since there is no restoring force at all in free fall. (v) If the pendulum is instead in a car accelerating horizontally with acceleration $a$, gravity and the horizontal pseudo-force combine vectorially …