How Is the Optimal Pivot Distance Calculated for a Uniform Disk Pendulum?

[Jimmy25]

Homework Statement

A uniform disk of radius R = 1.40 m and a 6.0 kg mass has a small hole a distance d from the disk's center that serves as a pivot point.

What should be the distance d so that this physical pendulum will have the shortest possible period?

Homework Equations

T = 2π[tex]\sqrt{\frac{I}{MgL}}[/tex]

The Attempt at a Solution

Using the parallel axis theorem and moment of inertia of a disk I found the period as:

T = 2π[tex]\sqrt{\frac{\frac{1.4^{2}}{2}+d^{2}}{9.8d}}[/tex]

When I find the minimum of this function I get d = 0.990 which is the right answer.

What I don't understand is why the d isn't zero. Won't T approach zero as d approaches zero?

 

[anathema]

it is important to understand that while T may approach zero as d approaches zero, it does not necessarily mean that d=0 will result in the shortest possible period. In this case, the period is also affected by the moment of inertia of the disk, which increases as d approaches zero. This means that even though the distance is decreasing, the overall effect on the period may not be significant enough to result in the shortest possible period. Additionally, there may be other factors at play that are not accounted for in the given equation, such as air resistance or friction, which could also affect the period. Therefore, it is important to carefully consider all variables and equations when determining the optimal distance for the shortest period.

 

[kerimek]

The distance d cannot be zero because that would mean the pivot point is at the center of the disk, which would result in no motion and therefore no period. In order for the physical pendulum to have a period, there must be a distance between the pivot point and the center of mass of the disk. As d approaches zero, the period does approach zero, but it never actually reaches zero. The minimum distance d = 0.990 ensures that the physical pendulum has the shortest possible period while still maintaining motion.