Compound pendulum clarification? I read in a book the following about compound pendulum and small displacements:  


*

*There are two points only for which the time period is minimum.  

*there are maximum 4 points for which the time period is same.  
Why is this? Can someone please explain? I am familiar with maximum time period being when $k=l$.
In general, time period is $$T=2\pi \sqrt\frac{k^2+l^2}{lg}$$ for small angle approximation.
$k$=Radius of gyration about the centre of gravity, $l$=distance of point of suspension from Centre of Gravity, $g$=gravity
 A: Hope you are familiar with differential calculus.
In order to get minimum time period differentiate time period w.r.t. $l$ and set it to $0$
Also check second derivative to be positive. That should give you $k=l$.
Note that locus of this will be a circle of radius $k$ and centre the centre of gravity.
Maybe your book is talking about a rod where there will be $2$ points only as other points are not lying on it.
For maximum time period : Clearly this will happen as $l$ tends to $0$ and time period tends to infinity.
For points where time period is same: Simply put time period at length $a$ to be equal to that of length $b$.
You will get $2$ solutions: $a=b$   or   $k^2 = ab$
As we want different points, we will ignore the first solution. There will be again infinite points satisfying this condition on a general body. But your textbook may be talking of a rod where there will be $4$ such points:
\begin{equation}
\begin{array}{cc}
a \; ,& -a \; ,& k^2/a \; ,& -k^2/a
\end{array}
\end{equation}
for a general non-zero a lying on rod.
