Finding Moment of Inertia: 2 Questions Answered

[jlmac2001]

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?

 

[Norman]

Originally posted by jlmac2001
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?

Start with:

$$\int r^2 dm = \int r^2 \sigma dA = \sigma \int r^2 dA$$

Where $\sigma$ is the constant density $\frac{M}{A}$

hope that helps

 

[jlmac2001]

dnn't understand

could someone explain itto me?

 

[Norman]

Originally posted by Norman
Start with:

$$\int r^2 dm = \int r^2 \sigma dA = \sigma \int r^2 dA$$

Where $\sigma$ is the constant density $\frac{M}{A}$

hope that helps

$$\sigma \int r^2 dA =\sigma \int (x^2+y^2)dxdy$$

this is ok since if you draw the rectangle out, r is measured from the center of the plane and therefore $r^2=x^2+y^2$. Since the center of the plane is the axis of rotation... you should be able to figure out the limits of integration for x and y.
Cheers.