calculating mi of ring whose axis passes through its diameter. please show thw steps.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Let us take a random plane shape (as in the first figure below) having 2 axis Ox1 and Ox2 in the plane of the figure and the Ox3 axis perpendicular to the plane of the figure (perpendicular to the paper).

For the moments of inertia of the shape in the figure about axis Ox1 and Ox2 we can write

`I_1 =int(y^2*dm)`

`I_2 =int(x^2*dm)`

As said, the third axis Ox3 is perpendicular to the plane of the figure, therefore

`I_3 = int(r^2*dm) =int(x^2+y^2)*dm = I_1+I_2`

Thus for a circular crown of mass m and radius R1 and R2 (see the second figure) we have

`I_1=I_2= (1/2)*I_3`

`I_3 =int(x^2+y^2)*dm`

and if we write

`sigma = m/S`

we have

`I_3 =int_(R1)^(R2)(sigma*r^2*2pir*dr) =(pi*sigma/2)*(R2^4-R1^4) =`


` `

` `

For a ring of radius R (=R1=R2) we have

`I_3 =I_z =m*R^2`

`I_1=I_2 (=I_x=I_y) =(1/2)*I_3 =(m*R^2)/2`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Approved by eNotes Editorial Team