When should the Parallel Axis Theorem be used in mechanics?

[amcavoy]

I know what the parallel axis theorem is, but I'm a little confused about when to use it. I recently had a problem where a hoop was rolling down an incline where I used the parallel axis theorem to find the translational acceleration and got it correct. However, I had a problem about a spool being pulled by a string (think of a yo-yo being pulled on the ground), and when I set up the equations I got the wrong answer using the P.A.T. For instance, I had:

$$F-F_{S}=ma$$

$$RF_{S}-rF=I_{CM}\alpha$$

Why isn't the I_CM instead I_P? I have solved the problem already and know the answer, I just can't see why the parallel axis theorem is not used. LINK: http://show.imagehosting.us/show/971155/0/nouser_971/T0_-1_971155.jpeg

Thank you for your help.

 

[Doc Al]

I just depends on what you take as your axis of rotation. If you take the center of mass, then I and torques will be about that point. (And you'll have no need for the parallel axis theorem.)

But you are certainly free to use the point of contact with the floor as your instantaneous axis of rotation. But if you do, be sure to take torques about that point as well. In this case you'll need to use the parallel axis theorem to find the rotational inertia about that point.

Done correctly, you'll get the same answer either way.

 

[quicknote]

The parallel axis theorem is a useful tool in mechanics that allows us to calculate the moment of inertia of an object about an axis parallel to its center of mass. This theorem is based on the understanding that the moment of inertia of an object is dependent on both its mass and its distance from the axis of rotation.

In the first problem you mentioned, the hoop rolling down an incline, the parallel axis theorem was applicable because the hoop was rotating about an axis parallel to its center of mass. However, in the second problem with the spool being pulled by a string, the spool is not rotating about an axis parallel to its center of mass. Instead, it is rotating about the point where the string is pulling on it, which is not parallel to the center of mass. This is why the moment of inertia about this axis is not equal to the moment of inertia about the center of mass, and the parallel axis theorem cannot be used.

In order to correctly solve this problem, you would need to use the general formula for moment of inertia, which takes into account the distance of each infinitesimal mass element from the axis of rotation. This would result in the moment of inertia about the point where the string is pulling on the spool, which is the correct value for this problem.

I hope this explanation helps clarify when to use the parallel axis theorem and when to use the general formula for moment of inertia. It's important to always carefully consider the axis of rotation and its relationship to the center of mass when applying this theorem. Keep up the good work in your problem-solving and scientific thinking!