Problem Understanding Physical Pendulum

[Saladsamurai]

I am having trouble understanding what h is in the equation for T listed below. Here is an image

The Problem: A physical pendulum consists of a uniform circular disk (of R=2.35cm) supported in a vertical plane by a pivot located a distance D=1.75cm from the center of the disk. The disk is displaced by a small angle and released. What is the period T of the resulting SHM?

Equations:
$$T=2\pi\sqrt{\frac{I}{mgh}}$$

$$I=I_{cm}+mH^2$$

from text $$I_{disk}=\frac{1}{2}MR^2$$

This is where I am messing up either my algebra or what I am interpreting as h.

For the parellel-axis theorem I had $$I_{disk}=\frac{MR^2}{2}+MD^2$$ where D is the distance from the COM to the pivot.

and for h in mgh I thought the total height was h=D+R.

Am I correct about what h is? Or no? This will determine what my mistake is

Thanks
Casey

 

[Saladsamurai]

Don't worry...I got it.

 

[m2003]

Dear Casey,

Thank you for reaching out regarding your trouble understanding the equation for T in the physical pendulum problem. I can see why you are confused about the value of h in the equation.

Firstly, h in the equation represents the height of the center of mass (COM) of the disk from the pivot point. This is different from the total height h=D+R, as you correctly mentioned. The reason for this is that in the equation T=2\pi\sqrt{\frac{I}{mgh}}, h is used to calculate the potential energy of the pendulum, which is directly proportional to the height of the COM from the pivot.

Secondly, your calculation of the moment of inertia (I) using the parallel-axis theorem is correct. However, in the equation I=I_{cm}+mH^2, H represents the distance from the pivot to the COM, not the disk diameter (D). This is why in your calculation, you used D as the distance to the pivot, but in the equation, H is used.

I hope this clarifies your doubts. If you are still having trouble understanding the problem, I would suggest breaking down the steps and equations into smaller parts and carefully considering the variables and their definitions. Additionally, it might be helpful to draw a diagram and label all the variables to visualize the problem better.

Best of luck with your problem-solving!

Sincerely,