Calculating Moment of Inertia of a Cuboid about the Z-Axis

In summary, Daniel explains that the moment of inertia for the cuboid depends on the axis of rotation. The calculation shown is correct for the moment of inertia about an axis through an edge, but for an axis through the center parallel to the z-axis, the MOI is (1/12)M(a^2 + b^2). The limits of integration also need to be adjusted.
  • #1
Nylex
552
2
How do I calculate this about the z-axis, if the cuboid length is b in the y-direction, a in the x-direction and c in the z -direction?

In my notes, I have I = ∫ r^2 dm = ∫ (x^2 + y^2) dm

dm = ρdV = ρdxdydz

This is what I did:

I = ∫∫∫ (x^2 + y^2)ρ dxdydz

I = ρ∫dz∫dy∫dx (x^2 + y^2)

I = ρ∫∫dy [(1/3)x^3 + xy^2] {0->a}

I = ρ∫∫[(1/3)a^3 + ay^2] dy

I = ρ∫dz [(1/3)ya^3 + (1/3)ay^3] {0->b}

I = ρ∫dz [(1/3)ba^3 + (1/3)ab^3]

I = ρ[(1/3)zba^3 + (1/3)zab^3] {0->c}

I = ρ[(1/3)cba^3 + (1/3)cab^3]

I = (1/3)ρabc(a^2 + b^2)

and ρabc = ρV = M, so I = (1/3)M(a^2 + b^2)

However, the answer has (1/12) instead of (1/3). Where does the other 1/4 come from??

Thanks.
 
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  • #2
The moment of inertia depends on the axis about which you want to rotate.
Your calculation is correct for the MOI about an axis that goes through the edge of the cuboid.

The MOI about an axis through its center and parallel with the z-axis is (1/12)M(a^2+b^2).
 
  • #3
The limits of integration need a little change...

Daniel.
 

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

The moment of inertia of a cuboid is a measure of its resistance to rotational motion around a specific axis. It is also known as rotational inertia and is represented by the symbol I. It is dependent on the mass distribution and the distance of the mass from the axis of rotation.

2. How is the moment of inertia of a cuboid calculated?

The moment of inertia of a cuboid can be calculated by using the formula I = (1/12) * m * (h^2 + w^2), where m is the mass of the cuboid and h and w are the height and width of the cuboid, respectively. This formula assumes that the axis of rotation passes through the center of mass of the cuboid.

3. How does the shape of a cuboid affect its moment of inertia?

The shape of a cuboid can greatly affect its moment of inertia. A cuboid with a larger height and width will have a higher moment of inertia compared to a cuboid with smaller dimensions and the same mass. This is because the mass is distributed further from the axis of rotation, making it harder to rotate.

4. Can the moment of inertia of a cuboid be changed?

Yes, the moment of inertia of a cuboid can be changed by altering its mass distribution or by changing the axis of rotation. For example, if the axis of rotation is moved closer to the center of mass, the moment of inertia will decrease, making it easier to rotate.

5. What is the significance of the moment of inertia of a cuboid?

The moment of inertia of a cuboid is an important concept in rotational dynamics and is used to calculate the angular acceleration of an object under the influence of a torque. It also plays a crucial role in understanding the behavior of objects in rotational motion and is essential in the design and analysis of machines and structures.

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