Rotational Inertia: Hoop vs Disk

[John1767]

Homework Statement

A ramp is set up on top of a table. An object is supposed to roll freely down the ramp off the table and have a horizontal displacement. Two objects are used in the same setup. One is a hollow hoop and another is a solid disk. The rotational inertia of the disk is higher than the rotational inertia of the hoop, their mass and radii are the same. Which object will have the larger horizontal displacement?

Relevant Equations

Krotational=1/2lw^2 Ktranslational=1/2mv^2 Ug=mgh Ihoop=MR^2 Idisk=1/2MR^2 tnet=I(alpha)

I know that a hoop should have a higher rotational inertia than a solid disk because its mass is distributed further from the axis of rotation. What I don't understand is how a disk of the same mass and radius can have a higher rotational inertia. If the objects roll freely their axes of rotation should be about their center so the equations I=MR^2 for the hoop and I=1/2MR^2 for the disk should apply. How can a disk somehow have a higher rotational inertia? I know that the disk should travel further than the hoop but this suggests the opposite.

 

[TSny]

Welcome to PF.

What I don't understand is how a disk of the same mass and radius can have a higher rotational inertia.

The hoop will have the larger rotational inertia, as the formulas show.

 

[hutchphd]

I agree: the statement of the problem seems contradictory to me. The smaller moment object (which should be the disc) indeed travels farther. Please check to see if somebody erred.

 

[TSny]

How can a disk somehow have a higher rotational inertia?

What makes you think that the disk would have a higher rotational inertia?

 

[phinds]

What makes you think that the disk would have a higher rotational inertia?

Uh ... maybe the fact the the problem statement says so (erroneously)?

 

[TSny]

Homework Statement:: ...The rotational inertia of the disk is higher than the rotational inertia of the hoop, their mass and radii are the same. ...

Ah, I missed this. This is clearly wrong.

 

[John1767]

The question itself says that Id > Ih where Id is the rotational inertia of the disk and Ih is the rotational inertial of the hoop.

 

[phinds]

The rotational inertia of the disk is higher than the rotational inertia of the hoop, their mass and radii are the same.

To quote Wally Shawn: "inconceivable!"

 

[TSny]

Did the problem statement actually state that the two objects have the same mass and radius? If not, then the disk could have a greater rotational than the hoop. I'm just looking at

 

[John1767]

The mass and radius of both the hoop and the disk are defined as m0 and r0 respectively, so maybe the question is giving me false information so I catch on to the false physics at work?

 

[TSny]

The mass and radius of both the hoop and the disk are defined as m0 and r0 respectively, so maybe the question is giving me false information so I catch on to the false physics at work?

OK. So, there is more to the problem statement than what is given in the figure that you posted in post #7.

Even if the disk had a greater mass and/or radius so that it had the greater rotational inertia, it would still travel the greater horizontal distance. So, it's not really important whether or not they have the same radius and mass.

 

[John1767]

Yes, this is the problem describing the hoop. It just defines the mass and radius with the same variables and describes the ramp situation.

 

[John1767]

So even though the disk has greater inertia, it will still have the larger horizontal displacement? Is there a concept that I'm missing or overlooked here?

 

[TSny]

Yes, this is the problem describing the hoop. It just defines the mass and radius with the same variables and describes the ramp situation.

OK, thanks. Yes, as @hutchphd said, there is an inconsistency in the problem statement. You were right to be puzzled about how $I_d$ could be greater than $I_h$ if the objects have the same mass and radius.

 

[John1767]

Alright, thank you. I'll just bring that up in my answer and hand it in. I appreciate the help.

 

[TSny]

So even though the disk has greater inertia, it will still have the larger horizontal displacement?

Yes. If the disk had a larger mass and/or radius so that it had a greater rotational inertia than the hoop, it would still travel farther horizontally.

Is there a concept that I'm missing or overlooked here?

Have you worked out an expression for the distance D for either the hoop or the disk? Does the result depend on either the radius or mass of the object?

 

[John1767]

Wow! I should have thought of that, sorry for the messy math but I just got so excited when I was able to cancel out mass. I guess the disk goes further regardless of somehow having more inertia, I guess it just threw me off that the disk had more inertia with the same mass and radius.

 

[TSny]

That looks right. (The time of flight will be the same for both objects.) Good!

 

[John1767]

Thanks for pointing me in the right direction!

 

[jbriggs444]

I guess the disk goes further regardless of somehow having more inertia

As has been pointed out, the disk does not have more rotational inertia in comparison to its mass. Instead it has less.

Ordinarily, rotational inertia and mass are not comparable. They do not share the same units. However, there is a meaningful way to compare them in this scenario by looking at rotational kinetic energy per unit mass for a rolling object at a given velocity versus ordinary translational kinetic energy for the same object at the same velocity.