Solving Metal Disk Problem: Find T

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In summary, a uniform metal disk with mass 8.21 kg and radius 1.88 m is free to oscillate as a physical pendulum about an axis through the edge. The formula for finding the period of small oscillations is T = 2π√(I/mgd), where I = mr²/2 + md² and d is the distance between the center and the edge of the disk. By using the parallel axis theorem, the correct formula is found to be T = 2π√(3r/g). Plug and chug can also be used to find the period.
  • #1
NasuSama
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Homework Statement



A uniform metal disk (M = 8.21 kg, R = 1.88 m) is free to oscillate as a physical pendulum about an axis through the edge. Find T, the period for small oscillations.

Homework Equations



[itex]I = mr^{2}/4[/itex]
[itex]T = 2\pi √(I/mgd)[/itex]

The Attempt at a Solution



I combined the formula together to get:

[itex]T = 2\pi √((mr^{2}/4)/(mgr))[/itex]
[itex]T = 2\pi √(r/(4g))[/itex]

But the answer is incorrect
 
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  • #2
NasuSama said:
[itex]I = mr^{2}/4[/itex]
How did you arrive at this result?
 
  • #3
Doc Al said:
How did you arrive at this result?

I am thinking that I need to use the moment of inertia of the disk.
 
  • #4
NasuSama said:
I am thinking that I need to use the moment of inertia of the disk.
Of course you do, but that's not the correct formula.
 
  • #5
Doc Al said:
Of course you do, but that's not the correct formula.

Then, it's something like I = mr²/2, rotating to its center. However, the disk oscillates through its edge.

I am not sure which path to go for...
 
  • #6
NasuSama said:
Then, it's something like I = mr²/2, rotating to its center.
Right.
However, the disk oscillates through its edge.
Use the parallel axis theorem. (Look it up!)
 
  • #7
Doc Al said:
Right.

Use the parallel axis theorem. (Look it up!)

Hm.. By the Parallel Axis Theorem, I would assume that:

[itex]I = I_{center} + md^{2}[/itex]
[itex]I = mr^{2}/2 + mr^{2}[/itex] [Since the disk rotates about an axis through the edge, we must add the inertia by mr². r is the distance between the center and the edge of the disk.]
[itex]I = 3mr^{2}/2[/itex]

Is that how I approach this? Let me know where I go wrong. Otherwise, I can just plug and chug this expression:

[itex]T = 2\pi √((3mr^{2}/2)/(mgr))[/itex]
[itex]T = 2\pi √(3r/(g))[/itex]
 
  • #8
Nvm. My answer is right. Thanks for your help by the way!
 
  • #9
Good! :approve:
 

1. What is the "Metal Disk Problem"?

The "Metal Disk Problem" is a theoretical problem in which a metal disk is heated and then allowed to cool down to room temperature. The goal is to determine the temperature at which the disk was heated (represented by T) based on the rate at which it cools down.

2. How do you solve the "Metal Disk Problem"?

To solve the "Metal Disk Problem", a scientist would use the principles of thermodynamics and heat transfer to create a mathematical model of the cooling process. This model would then be used to calculate the value of T, the unknown temperature at which the disk was originally heated.

3. What factors affect the rate of cooling in the "Metal Disk Problem"?

The rate of cooling in the "Metal Disk Problem" is affected by various factors such as the material and thickness of the disk, the ambient temperature, and the rate of heat transfer from the disk to the surrounding environment.

4. Can the "Metal Disk Problem" be solved using experimental methods?

Yes, the "Metal Disk Problem" can be solved using experimental methods. However, they may not provide an exact solution and can be time-consuming and costly. Therefore, mathematical modeling and simulations are often used to solve this problem.

5. What are the practical applications of solving the "Metal Disk Problem"?

Solving the "Metal Disk Problem" has practical applications in various industries such as metallurgy, manufacturing, and materials science. It can also help in understanding the principles of heat transfer and thermodynamics, which are crucial in many engineering and scientific fields.

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