What is the Moment of Inertia for a Solid Door?

[howsockgothap]

Homework Statement

A solid door of mass 25.40 kg is 2.31 m high, 1.48 m wide, and 2.58 cm thick. What is the moment of inertia of the door about the axis through its hinges?

Homework Equations

I=(1/12)m(a^2 + b^2) + mr^2

(parallel axis theorem)

The Attempt at a Solution

I initially used the door height and width as a and b (I'm sure this is right) and used the door thickness as the distance between axis. On further thought it's obvious that my distance between the axis (r) is not the door thickness. I guess my confusion comes from knowing what to use in that area of the eq'n.

 

[kuruman]

The relevant equation that you have put down is for a door of mass m but infinitely thin, i.e. two-dimensional. How do you think you should modify the equation if the door has finite thickness, i.e. is three-dimensional?

 

[howsockgothap]

The only solution I can determine to that is using several different equations for moment of inertia with height/width, width/thickness and thickness/height in the place of a and b. However I don't understand how I can then utilize those different equations with the parallel axis theorem.

 

[kuruman]

Can you find an expression for the moment of inertia of a three-dimensional about an axis that goes through the CM and is parallel to the long (vertical) axis and perpendicular to the other two axes? If yes, then you can use the parallel axes theorem to find the moment of inertia about the hinge.

 

[howsockgothap]

aaahhhh... 1/3(m)a^2

embarrassingly simple, in hindsight. Thanks.

 

[kuruman]

Not quite. That would be the case if the door had no thickness. What is the moment of inertia of a rectangle of length a=1.48 m and width b=2.48 cm about its end? That's what you want.