Moment of Inertia | Solve a Problem with Centroid & Steiner's Theorem

In summary, the individual is trying to solve a problem for which they are unfamiliar and have had difficulty with. They state that they know how to solve the problem, but have had difficulties in getting the correct results. They mention that they are using Steiner's theorem to calculate the moment of inertia, but do not provide any further details. They provide a file which may or may not be helpful in understanding the problem.
  • #1
HERiTAGE
5
0
Hey ppl it's my first post wohooo. :)

I've been trying o solve this problem for ages and seriously i don't know what I'm doing wrong... It's not a difficult problem but i can't seem to solve it.
I have to calculate the MoI of this figure:

http://mypicfordc.com.sapo.pt/moi.JPG

Where a=89.9 [cm], b=57.8 [cm], c=44.8 [cm]

--------------------------------------------------------------------------------------------------------

I've been able to accuraely calculate the centroid of the figure (xG=50.261 ; yG= 44.607) and I've reached a MoI value that is close to the solution but it's not close enough to be considered correct.

My MoI values are:
Ix= 5305338 ; Iy= 4131811 ; Ixy= 1580647

Correct values are surrounding:
Ix= 5800000 ; Iy= 4700000 ; Ixy= 1500000 (not completely accurate, just an aprox.) The main values are all correct (Areas, xGi, yGi, Sxi, Syi) because i need the to find the Centroid, which is correct.
I'm using Steiner's theorem to find the moment of inertia.

Please help me, I'm driving nutts...

I can scan my calculus so you can see what i am doing wrong but it won't do you much because it's kind of messy. (if you still would like to see it, just ask)

Thanks in advance. :)
 
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  • #2
Does Steiner's formula compute the inertia using the double integral?

(I'm not familiar with it at all!)

Assuming it does, I don't see how, if you computed the centroid using double integration, that your moments of inertia can be wrong.

Perhaps you could put up extended workings?
 
  • #3
No, no integration is required since the shapes that compose the figure are pretty common. (quarter circle, rectangle and triangle)

It's kind of hard for me to explain how Steiner Theorem works through a keyboard and speaking in another language than my natural one... :)

What Steiner theorem does is, calculate the Moment of Inertia of a figure considering a given point.

The formula for the I'x (i.e.) is:

I'x= Ix+dy*A

Where Ix is the Moment of Inertia of your figure (there are tables that give you formulas for Moments of Inertia on squares, circles, etc) dy is the distance between your figure's centroid and the main figure's centroid and A is the Area of your figure.

example using my figure:

The moment of inertia on a quarter circle is giver by:

Ix=(pi*r^4)/16

So using Steiner's theorem to calculate the MoI of the Quarter circle on the main figure's centroid we get:

Ix'=(pi*r^4)/16 + dy * (pi*r^2)/4

where dy is the difference between yG of the quarter circle and yG of the main figure.
 
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  • #5
Well can anyone help? please?
 
  • #6
If you have the MOI of each piece (quarter circle, triangle, square) about its COM, use the parallel axis theorem to find the MOI of each about the centroid. Then just add them up. Of course, the masses of the missing pieces of the square will be negative.
 
  • #7
I've done that.
That's how i got that result.

The thing is, i know how to solve the problem, but i keep getting a wrong result.
I even checked the problem step by step with a friend of mine and we both got to the conclusion that the methodology was correct. Only the results weren't. :\

I really need to solve this problem coz it's the final of a series of 20 problems that i must solve in order to go to the final exam and it would suck to flunk a discipline just because i couldn't solve one problem.
 
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  • #8
HERiTAGE said:
What Steiner theorem does is, calculate the Moment of Inertia of a figure considering a given point.

The formula for the I'x (i.e.) is:

I'x= Ix+dy*A

Where Ix is the Moment of Inertia of your figure (there are tables that give you formulas for Moments of Inertia on squares, circles, etc) dy is the distance between your figure's centroid and the main figure's centroid and A is the Area of your figure.
Not clear to me what you are doing here. As has been suggested, use the parallel axis theorem. Using your notation, that would be:
I'x= Ix+ M(dy)^2, where M is the mass of the piece. (See the link that J77 provided.)
 
  • #9
It's hard for me to explain what I'm trying to do.
I am using the parallel axis theorem (a.k.a. Steiner's Theorem).

Hmm maybe i should upload my scanned calculus...

http://mypicfordc.com.sapo.pt/moical.jpg"

*Forget about the lower right values for a, b and c. The correct ones are near the figure.

*Edit*

Never mind... I've already solved the problem. I had a mistake on on of the tables where i calculate the centroid of each composing figure.
Thanks for the help anyway.
 
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1. What is moment of inertia?

Moment of inertia, also known as rotational inertia, is a physical property of an object that describes its resistance to changes in its rotational motion. It depends on the mass and distribution of mass of the object.

2. How is moment of inertia calculated?

Moment of inertia is calculated by integrating the product of mass and squared distance from the axis of rotation over the entire object. The formula for moment of inertia is I = ∫r²dm, where I is the moment of inertia, r is the distance from the axis of rotation, and dm is the mass element.

3. What is the significance of centroid in moment of inertia?

The centroid is the geometric center of an object, and it plays a crucial role in calculating moment of inertia. The moment of inertia of an object can be simplified by using the centroid as the reference point, making calculations easier.

4. What is Steiner's theorem and how is it used in solving problems related to moment of inertia?

Steiner's theorem states that the moment of inertia of an object about a certain axis can be calculated by adding the moment of inertia of the object about its centroid (IC) with the product of the object's mass (m) and the squared distance between the given axis and the centroid (d²). This theorem is used to simplify the calculation of moment of inertia for complex objects.

5. What are some real-life applications of moment of inertia?

Moment of inertia has numerous applications in real life, such as in the design of rotating machinery, such as turbines and engines. It is also used in the design of vehicles, such as cars and airplanes, to ensure stability and control. In sports, moment of inertia is important in activities like figure skating and gymnastics, where rotational movements are involved.

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