Explaining Mass Moment of Inertia: Izz Different from Ixx and Iyy?

In summary: Now I feel much better.In summary, someone asks for help calculating MOI for a rectangular block, but provides incorrect formulas. The formulas provided by the person are correct, but the x- and y- axes don't pass through the center of the COM, making the mass moment of inertia about the z-axis different.
  • #1
Fermat
Homework Helper
872
1
Can someone explain something to me ?

I refer you to http://www.engineering.com/content/ContentDisplay?contentId=41005050".

If we consider the rectangular block to be a cube, Then I would expect that we would get the mass moments of inertia all equal, Ixx = Iyy = Izz.
But, in the page I referred you to, if we let a = b = L, then we get Ixx = Iyy, but Izz is different.

Can anyone explain why Izz is different from Ixx and Iyy ?
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Good question. Looks wrong to me too.
 
  • #3
Thanks. Now I don't feel so bad.

Someone had asked me for help in calculatring MOI for a rectangular block.

I did some working and got the Izz format in that page. And posted my working on the forum I was at.

Then I decided to check it out on the web.

Imagine my consternation when I found that web page !

I hope some more people tell me it is wrong <hopes mightily>
 
  • #4
Gottit!

The x-axis and the y-axis don't pass through the COM of the block, but the z-axis does.

Ixcxc is of the same form as Izz, but Ixx needs the parallel axes theorem giving,

Ixx = Ixcxc + M(L/2)²

which gives,

Ixx = (1/12)Ma² + (1/3)ML²

So, my working was right after all :smile:

Oh, yes. And that web page is also correct.
 
  • #5
Are you sure you looked at the right fomulas?

You had to observe x_C, y_c,z_c, NOT X,y,Z.

x_c,y_c,z_c ARE equal.

(You are right in your observation abotu x and y not being in the center.)

Be aware that the COM is locationally dependent!
 
Last edited:

FAQ: Explaining Mass Moment of Inertia: Izz Different from Ixx and Iyy?

What is mass moment of inertia?

Mass moment of inertia is a property of a physical object that describes its resistance to rotational motion. It is the measure of an object's distribution of mass around a given axis of rotation.

How is Izz different from Ixx and Iyy?

Izz, Ixx, and Iyy are the three principal moments of inertia. Izz represents the moment of inertia about the object's axis of rotation, while Ixx and Iyy represent the moments of inertia about two perpendicular axes. The values of Ixx and Iyy are usually different from each other and from Izz because they are calculated based on the object's shape and orientation.

How is mass moment of inertia calculated?

Mass moment of inertia can be calculated using the formula I = ∫ r^2 dm, where r is the distance from the axis of rotation to each element of mass, and dm is the mass of each element. This integral is typically solved using calculus methods.

Why is mass moment of inertia important?

Mass moment of inertia is important because it affects an object's ability to resist rotational motion. Objects with a higher mass moment of inertia will be more difficult to rotate and will require more torque to do so. This property is crucial in fields such as engineering and physics, where understanding rotational motion is necessary.

How does mass moment of inertia impact everyday objects?

Mass moment of inertia impacts everyday objects in many ways. For example, it affects how cars handle and maneuver, how buildings resist wind forces, and how athletes perform gymnastic routines. It is also important in the design of tools and machinery, as it determines how much force is needed to rotate or stop a rotating object.

Similar threads

Back
Top