Kinetic Energy Ratio of an Eccentric Disk

[mc120]

1. Let’s say we have a solid wheel. The wheel can be modeled as a disk. Imagine that instead, the wheel is rotated at a location location 0.47R from the center of the wheel, so that the wheel rolled around a kind of loop. Essentially, the CM goes around the dashed line in the drawing. R is the radius. What is the percentage of the total kinetic energy that must be rotational?

2. I am guessing at these being relevant:
I_disk=1/2mr²
I_total=1/2mr²+md²
KE_rot=1/2Iω²
KE_trans=1/2mv²


3. Ratio = KE_rot/(KE_rot+KE_trans)
Substituting in: I_total=1/2mr²+m(0.47r)² into KE_rot=1/2Iω² and KE_trans=1/2mω² (since the CM will travel 2∏r in the same time period as rotation), I simplify to:
(1/2mr²+1/2m(0.47r)²)/(1/2mr²+1/2m(0.47r)²+m), which is the same as:
(1/2r²+(0.47r)²)/(1/2r²+(0.47r)²), which equals 1.


I think I have gone horribly awry in my assumptions. I'm not really sure I even understand why there is anything OTHER than rotational Kinetic energy in this equation. The parallel axis theory makes sense for calculating the Inertia of an eccentric disk, but isn't it all still rotational kinetic energy?

Thanks for any advice anyone can give. Cheers!

 

[SammyS]

[ b]1. Let’s say we have a solid wheel. The wheel can be modeled as a disk. Imagine that instead, the wheel is rotated at a location location 0.47R from the center of the wheel, so that the wheel rolled around a kind of loop. Essentially, the CM goes around the dashed line in the drawing. R is the radius. What is the percentage of the total kinetic energy that must be rotational?

[/b]

[ b]2. I am guessing at these being relevant:
I_disk=1/2mr²
I_total=1/2mr²+md²
KE_rot=1/2Iω²
KE_trans=1/2mv²
[/b]

[ b]3. Ratio = KE_rot/(KE_rot+KE_trans)
Substituting in: I_total=1/2mr²+m(0.47r)² into KE_rot=1/2Iω² and KE_trans=1/2mω² (since the CM will travel 2∏r in the same time period as rotation), I simplify to:
(1/2mr²+1/2m(0.47r)²)/(1/2mr²+1/2m(0.47r)²+m), which is the same as:
(1/2r²+(0.47r)²)/(1/2r²+(0.47r)²), which equals 1.
[/b]

I think I have gone horribly awry in my assumptions. I'm not really sure I even understand why there is anything OTHER than rotational Kinetic energy in this equation. The parallel axis theory makes sense for calculating the Inertia of an eccentric disk, but isn't it all still rotational kinetic energy?

Thanksfor any advice anyone can give. Cheers!

Hello mc120. Welcome to PF !

Although, you could consider all the KE to be rotational, I think that the key here is the word must .

What is the percentage of the total kinetic energy that must be rotational?

(Please don't use bold excessively.)

 

[mc120]

Thanks for the reply SammyS.

I'm sorry, I typed within the [ b ] tags thinking they were the "question template" per the sticky FAQ thread. Oops!

I understand that I need to figure out what part of the energy is translational and which is rotational, but my equations are not really working out for me. Is it right to think of translational energy in this problem as the part of the eccentric disk where the Centre of Mass is traveling an arc length?

 

[SammyS]

Thanks for the reply SammyS.

I'm sorry, I typed within the [ b ] tags thinking they were the "question template" per the sticky FAQ thread. Oops!

I understand that I need to figure out what part of the energy is translational and which is rotational, but my equations are not really working out for me. Is it right to think of translational energy in this problem as the part of the eccentric disk where the Centre of Mass is traveling an arc length?

That seems right to me.