Period of a physical pendulum with a pivot at its center

[PhysicsKush]

Homework Statement

The picture illustrates a simple pendulum and and two physical pendulums ,all having the same length ,L. Class their period in ascending order.

Homework Equations

T = 2π / ( I/mgh)
I = I_cm + mh²

I_cm=(ML²/12)

The Attempt at a Solution

I have found the period for first two objects :
T₁ = 2π√(L/g)
T₂ = 2π√(2L/3g)

The problem for me appears at the third system because h is defined as the distance between the center of mass of the rod and the pivot point. In the third situation, it seems to me that the pivot is at the center of mass , hence h=0 , but that would result in a division by 0 error.

I have tried cutting the rod in half so that L =L/2 and h = L/4 but after some algebra i end up with T₃ = 2π√(L/3g) , which is smaller than T₁

The solution to the problem is T₃ > T₁ > T₂

Any help would be apreciated.

 

[kuruman]

The problem for me appears at the third system because h is defined as the distance between the center of mass of the rod and the pivot point. In the third situation, it seems to me that the pivot is at the center of mass , hence h=0 , but that would result in a division by 0 error.

And what do you get when you have a fraction the denominator of which is very small but not exactly zero? How does the period compare with the other two periods? Will the ranking of the periods change when the denominator is exactly zero?

 

[PhysicsKush]

And what do you get when you have a fraction the denominator of which is very small but not exactly zero? How does the period compare with the other two periods? Will the ranking of the periods change when the denominator is exactly zero?

Not sure what you mean :/

 

[kuruman]

Not sure what you mean :/

I mean: You said the period is $T = 2π\sqrt{I/(mgh)}$. That's good. What is $h$ in case (iii)? You said $h$ is zero. That is correct, but you seem to be stuck at the mathematical impossibility of dividing by zero. Mathematically, $\frac{1}{0}=\infty$. We are doing physics here where there are hardly any mathematical infinities. How do you physically interpret "infinite period"? Answer: Very long. OK, if T₃ is just "very long" as opposed to "very, very, very long", how do you think that would affect its ranking against T₁ and T₂? In other words, if one quantity is infinite or nearly infinite, could it be smaller than two finite quantities?