Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.