How to use parallel axis theorem?

[asz304]

Homework Statement

A solid door of mass 37.80 kg is 2.30 m high, 1.70 m wide, and 2.53 cm thick. What is the moment of inertia of the door about the axis through its hinges?

Homework Equations

I= Icm + MD^2
Icm = 1/12[M (a^2 +b^2 )] formula for inertia of a rectangular plate.

The Attempt at a Solution

so I found Icm = 25.767 kgm^2 by doing Icm = 1/12 [ 37.80 kg ( 1.7^2 + 2.3^2 ) ].

and I'm stuck in doing the parallel axis theorem, and I'm not sure if my work above is right.

 

[Filip Larsen]

You need to find the correct CM axis to take the MOI around first (hint: the axis for I_CM must be parallel to the final axis. You seem to have found an orthogonal axis instead).

 

[asz304]

I can't get what you're saying..

 

[Filip Larsen]

The parallel axis theorem states that the moment of inertia (MOI) around any axis (which would be your door hinge axis in your case) can be found by taking the MOI around a parallel axis that goes through the center of mass (CM) of the body (and then add the term you have written as MD²).

In calculating I_CM the equation you used indicates that you have chosen an axis of rotation that is perpendicular to the height and width of the door, hence an axis that is not parallel to the hinge axis. And if it is not parallel you cannot apply the parallel axis theorem. Thus, you need to find I_CM around a different axis that is parallel to the hinge axis.

Alternatively, you can find I_hinge directly. Using the parallel axis theorem only makes sense if find I_CM is easier than finding I_hinge.

 

[kudoushinichi88]

If you think about it, a door rotating about its hinges is like a rod rotating about one of its ends. 8D