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

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

  2. 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

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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.

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