Maximum points that giv esame time period for compound pendulum
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Hope you are familiar with differential calculus.
In order to get minimum time period differentiate time period w.r.t. ll and set it to 00 Also check second derivative to be positive. That should give you k=lk=l.
Note that locus of this will be a circle of radius kk and centre the centre of gravity. Maybe your book is talking about a rod where there will be 22 points only as other points are not lying on it.
For maximum time period : Clearly this will happen as lltends to 00 and time period tends to infinity.
For points where time period is same: Simply put time period at length aa to be equal to that of length bb.
You will get 22 solutions: a=ba=b ork2=abk2=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 44 such points:
a,−a,k2/a,−k2/aa,−a,k2/a,−k2/a
for a general non-zero a lying on rod.
In order to get minimum time period differentiate time period w.r.t. ll and set it to 00 Also check second derivative to be positive. That should give you k=lk=l.
Note that locus of this will be a circle of radius kk and centre the centre of gravity. Maybe your book is talking about a rod where there will be 22 points only as other points are not lying on it.
For maximum time period : Clearly this will happen as lltends to 00 and time period tends to infinity.
For points where time period is same: Simply put time period at length aa to be equal to that of length bb.
You will get 22 solutions: a=ba=b ork2=abk2=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 44 such points:
a,−a,k2/a,−k2/aa,−a,k2/a,−k2/a
for a general non-zero a lying on rod.
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