Q. Consider a physical pendulum and a simple pendulum both of length $L$ . The physical pendulum is a thin rigid and uniform rod hinged to the ceiling so that it is free to swing back and forth about its one end. Now for a small oscillation, find the ratio of time periods of the physical pendulum and the simple pendulum:
NTA AbhyasNTA Abhyas 2020
Solution:
In case of rod we will consider it a compound pendulum.
the time period of physical pendulum will be
$T_{A}=2\pi \sqrt{\frac{I_{o}}{m g l}}$
Io= moment of inertia of rod
l= length of rod.
$T_{A}=2\pi \sqrt{\frac{\frac{m l^{2}}{3}}{\frac{m g l}{2}}}=2\pi \sqrt{\frac{2 l}{3 g}}$ .......(1)
for simple pendulum
and $T_{n}=2\pi \sqrt{\frac{l}{g}}\Rightarrow \frac{T_{A}}{T_{b}}=\sqrt{\frac{2}{3}}$ .......(2)
the time period of physical pendulum will be
$T_{A}=2\pi \sqrt{\frac{I_{o}}{m g l}}$
Io= moment of inertia of rod
l= length of rod.
$T_{A}=2\pi \sqrt{\frac{\frac{m l^{2}}{3}}{\frac{m g l}{2}}}=2\pi \sqrt{\frac{2 l}{3 g}}$ .......(1)
for simple pendulum
and $T_{n}=2\pi \sqrt{\frac{l}{g}}\Rightarrow \frac{T_{A}}{T_{b}}=\sqrt{\frac{2}{3}}$ .......(2)