Why Does the Period of a Pendulum Increase with Distance?

[mit_hacker]

Homework Statement

A solid, uniform disk of mass M and radius a may be rotated about any axis parallel to the disk axis, at variable distances from the center of the disk.

If you use this disk as a pendulum bob, what is T(d), the period of the pendulum, if the axis is a distance d from the center of mass of the disk?

I got the answer to be:

T(d) =2{\pi}\sqrt{\frac{0.5a^{2}+d^{2}}{gd}}

The period of the pendulum has an extremum (a local maximum or a local minimum) for some value of d between zero and infinity. Is it a local maximum or a local minimum?

Homework Equations

The Attempt at a Solution

I graphed the function and saw that as d increases, the period T also increases to a maximum. However, I am not clear about the physical reason as to why this is so. Please advise.

Thank-you!

 

[anathema]

Thank you for your post. Your solution for the period T(d) of the pendulum using a solid, uniform disk is correct. As for your question regarding the extremum of the period, it is a local minimum. This means that for a specific value of d, the period T will be at its minimum value. This can be seen from the equation you provided, where as d increases, the denominator (gd) also increases, resulting in a smaller value for the overall fraction and thus a smaller period T.

The physical reason for this is due to the moment of inertia of the disk. As the distance d increases, the disk's moment of inertia also increases, making it more difficult for the pendulum to swing back and forth. This results in a longer period T. However, at a certain distance d, the moment of inertia reaches its maximum value and any further increase in d will result in a decrease in the period T. This is why the period T has a local minimum at a specific distance d.

I hope this helps clarify your understanding. Keep up the good work in your studies!

 

[ogb p]

The physical reason for this is due to the moment of inertia of the disk. As the distance d increases, the moment of inertia also increases, leading to a longer period of oscillation. This is because the larger moment of inertia requires more energy to overcome and complete a full oscillation. Therefore, the period T increases until it reaches a maximum and then starts to decrease again as d continues to increase. This can also be seen in the equation, where as d increases, the denominator also increases, resulting in a longer period T.