Relate rotational kinetic energy to potential energy

[lydia_y620]

Homework Statement

This problem is from the 2015 AP Physics C Mechanics free response, question 3 part b.
https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap15_frq_physics_c-m.pdf

Homework Equations

K = 1/2Iω²
U = mgh

The Attempt at a Solution

The potential energy of the bar when it's horizontal gets transferred to kinetic energy when vertical, so I originally had the equation mgL = 1/2(1/3ML²)ω²
However, the scoring guidelines say that the change in height should be L/2, not L, resulting in a potential energy of mgL/2. Could someone explain why the change in height should be L/2 and not L?

 

[vela]

The change in vertical position of the end of the bar is L. Is that the point that's important in the problem?

 

[lydia_y620]

The change in vertical position of the end of the bar is L. Is that the point that's important in the problem?

It asks for the velocity of the "free end of the rod", so I think that it means the end of the bar. But why does the answer sheet say L/2 then?

 

[vela]

Is that relevant to the change in potential energy of the rod? Why not use the position of the other end or a point 1/3 the way in from one end? I'm trying to get you to think about what point is important when you talk about the potential energy of the rod.

 

[lydia_y620]

Is that relevant to the change in potential energy of the rod? Why not use the position of the other end or a point 1/3 the way in from one end? I'm trying to get you to think about what point is important when you talk about the potential energy of the rod.

So I have to take the entire bar into account and not just the end...kind of like taking the "average" change in height for all pieces of the rod?

 

[vela]

Yes. More precisely, the average position of the mass of the bar, i.e., the center of mass.

 

[lydia_y620]

Yes. More precisely, the average position of the mass of the bar, i.e., the center of mass.

Okay, that makes sense. Thanks for the help!