Applying the parallel axis theorem to find inertia

[vetgirl1990]

Homework Statement

Calculate the moment of inertia of a uniform rigid rod of length L and mass M, about an axis perpendicular to the rod through one end.

Homework Equations

Parallel axis theorem: I = Icm + MD²
Long thin rod with rotation axis through centre: Icm = 1/12 ML²
Long thin rod with rotation axis through end: I = 1/3 ML²

The Attempt at a Solution

I know this is a straightforward substitution problem into the parallel axis theorem EQN.
However, for this question, I'm not sure why the answer key uses Icm of the rotation axis through the CENTRE. The question specifically states that the rotation axis is at the end...

The answer given is I = 1/3ML²;
I have calculated I = 7/12ML², using the Icm of a long thin rod with the rotation axis through the end.
Calculations:
D = 1/2L, since the centre of mass of a rod is right down the middle
I = Icm + MD²
I = 1/3 ML² + M(1/2L)²
I = 7/12ML²

To reiterate, I'm just confused about why Icm was used for am axis through the centre, rather than through the end.

 

[azizlwl]

Can you show your calculation?

 

[vetgirl1990]

Can you show your calculation?

Just edited my post with the full calculations.

 

[azizlwl]

Why I_{cm[/SUB=1/3ML². Different from you stated in relevant equations.}

 

[vetgirl1990]

Why I_{cm[/SUB=1/3ML². Different from you stated in relevant equations.}

I'm not sure which part of the solution you're referring to.
I = Icm + MD² is the parallel axis theory, and I plugged in I = 1/3 ML² the Icm component of that equation.

 

[haruspex]

I'm not sure which part of the solution you're referring to.
I = Icm + MD² is the parallel axis theory, and I plugged in I = 1/3 ML² the Icm component of that equation.

You seem to be applying the parallel axis theorem twice over.
The MoI about the rod's centre, axis perpendicular to the rod, is ML²/12. By the parallel axis theorem (or otherwise) the MoI about one end is ML²/12+M(L/2)²=ML²/3. You quoted this as one of your known equations, and it is directly the answer to the question. There is no need to go adding another ML²/4.
Note that the Icm in the parallel axis theorem is the MoI for an axis through the object's mass centre. It is not valid to substitute in there the MoI about any other axis.