What is the Moment of Inertia for a Solid Door?

In summary, the conversation discusses the calculation of the moment of inertia of a solid door with given dimensions. The relevant equation is modified for a three-dimensional object, and the parallel axis theorem is used to find the moment of inertia about the hinge. The final equation for the moment of inertia is found to be 1/3(m)a^2.
  • #1
howsockgothap
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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.
 
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  • #2
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?
 
  • #3
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.
 
  • #4
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.
 
  • #5
aaahhhh... 1/3(m)a^2

embarrassingly simple, in hindsight. Thanks.
 
  • #6
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.
 

1. What is the moment of inertia of a door?

The moment of inertia of a door is a measurement of its resistance to rotational motion around a specific axis. It is influenced by the mass, shape, and distribution of the door's material.

2. How is the moment of inertia calculated for a door?

The moment of inertia of a door can be calculated using the door's mass, dimensions, and the parallel axis theorem. It is typically expressed in units of kilograms per square meter (kg/m^2).

3. Why is the moment of inertia important for a door?

The moment of inertia is important for a door because it affects how easily the door can be opened or closed. A higher moment of inertia means it will take more force to rotate the door, while a lower moment of inertia means it will take less force.

4. How does the moment of inertia change when a door is opened or closed?

The moment of inertia of a door changes as it is opened or closed because the door's mass distribution changes. For example, when a door is opened, more of its mass is located farther from the rotational axis, increasing its moment of inertia.

5. How can the moment of inertia of a door be adjusted?

The moment of inertia of a door can be adjusted by changing the door's mass distribution. This can be done by adding or removing weight from different areas of the door, such as the hinges or the handle. Altering the door's shape can also affect its moment of inertia.

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