How do I find the moment of inertia for shapes with uniform density?

In summary, the conversation discusses two questions related to finding the moment of inertia of different shapes. The first question involves a rectangular sheet with uniform density, where the formula I = (integral over A)M/A dA is used and the limits of integration are determined by the shape and size of the sheet. The second question involves a thin uniform disk, where the formula I = 2M/a^2 (a^4/4) is used and the correct answer is M(a^2/48+b^2/48). The conversation also mentions using double integrals and shows the correct limits of integration for the first question.
  • #1
jlmac2001
75
0
I'm don't really know how to find the momemt of inertia. I have two questions that I'm stuck on.

Two questions:

1. Find the moment of inertia of a sheet f mass M and uniform density which is in the shape of a rectangle of sides a and b, for rotations about an axis passing through its center and perpendicular to the sheet.

answer:Will I start with this I= (integral over A)M/A dA? How would I find the limits of integration and integrate this?

2. Find the moment of inertia of a thin uniform disk of mass M and radius a for rotations about an axis through a diameter of the disk.

answer: Will th answer be I=2M/a^2 (a^4/4)=Ma^2/2?
 
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  • #2
1. that isn't the answer, no

I is the integral over A of pr^2 dA where p is the density and r is the distance from the origin


ie int over A of p(x^2+y^2)dxdy

the limits for x are -a/2 to a/2 and y is -b/2 to b/2

also use abp=M, I'm presuming you can do double integrals - this one is quite easy.

the second one is somewhat harder, but just try this for now.
 
  • #3
is this right

M(a^2/48+b^2/48) is the moment of inertia
 
  • #4
you're about a factor of 16 out, how did you get that? show your working...
 

FAQ: How do I find the moment of inertia for shapes with uniform density?

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is often described as the rotational equivalent of mass.

How is moment of inertia calculated?

The moment of inertia of an object can be calculated by multiplying the mass of the object by the square of its distance from the axis of rotation.

Why is moment of inertia important?

Moment of inertia is important because it helps us understand how objects will behave when subjected to rotational forces. It is crucial in many areas of physics and engineering, such as designing rotating machinery and predicting the motion of objects in space.

How does moment of inertia affect an object's motion?

Moment of inertia affects an object's motion by determining how difficult it is to change its rotational motion. Objects with a larger moment of inertia will be harder to rotate, while those with a smaller moment of inertia will be easier to rotate.

Can moment of inertia be changed?

Yes, moment of inertia can be changed by altering an object's mass or its distribution of mass. For example, increasing the mass of an object or moving the mass closer to the axis of rotation will decrease its moment of inertia.

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