Physics Homework Help With Moment Of Inertia Of Continuous Body

Moment Of Inertia Of Continuous Body

For calculating moment of inertia of a continuous body, we first divide the body into suitably chosen infinitesimal elements. The choice depends on symmetry of body. Consider an element of the body at a distance r from the axis of rotation. The moment of inertia of this element about the axis we defined as (dm)r2 and the discrete sum over particles becomes integral over the body:

moment of inertia of continuous body image

moment of inertia of continuous body image1 … (vii)

Moment of Inertia for different bodies

1. Moment of inertia of a rod

(a) About an axis passing through its centre and perpendicular to its length. Considering a small element of the rod of length dx and mass dm at a distance x from the axis where moment of inertia of continuous body image3

The moment of inertia of this element about the given axis is dI = dm.x2 = moment of inertia of continuous body image4

Hence the moment of inertia of whole rod is

I = moment of inertia of continuous body image5

(b) About an axis touching the edge of the rod and perpendicular to the length of the rod.

It can be obtained by using the limits from zero to L

where moment of inertia of continuous body image6

moment of inertia of continuous body image7

I = moment of inertia of continuous body image8

2. Uniform Circular ring:

If M is the mass of the ring with radius R, then moment of inertia about an axis passing through the center of ring perpendicular to its plane can be obtained by

moment of inertia of continuous body image9 moment-of inertia of continuous body image10

3. Uniform circular Disc:

Uniform circular disc about an axis passing through its centre and perpendicular to its plane. Dividing the disc into a number of concentric rings and considering one ring of radius r and thickness dr. Let due to the mass of the disc and moment of inertia of continuous body image11

(i) given by moment of-inertia-of continuous-body image12

The moment of inertia about the given axis of this ring

moment of inertia of continuous body image13 therefore the moment of inertia of the whole disc is given by moment of inertia of continuous body image14

It may be noted that the moment of inertia of a disc and of a ring is independent of their thickness.

Moment of Inertia of Some Common Shaped Bodies

A Thin Rod

The axis is perpendicular to the rod and passing through its centre.

Ic = moment of inertia of continuous body image15
moment of inertia of continuous body image16 A Thick Rod

The axis is perpendicular to the rod and passing through the centre.

moment of inertia of continuous body image17
moment of inertia of continuous body image18
A Ring

Axis is perpendicular to the plane of the ring

moment of inertia of continuous body image19
moment of inertia of continuous body image20 A Hollow Cylinder

moment of inertia of continuous body image25
moment of inertia of continuous body image22
A Disc

The axis perpendicular to the plane of the disc.

moment of inertia of continuous body image23
moment of inertia of continuous body image24 A Solid Cylinder

moment of inertia of continuous body image25
moment of inertia of continuous body image26
A thin rod about a perpendicular axis through its end.

I = moment of inertia of continuous body image27
moment of inertia of continuous body image28 A rectangular plate about one edge.

moment of inertia of continuous body image29
moment of inertia of continuous body image30
A Solid Sphere

About its diameter

moment of inertia of continuous body image31
moment of inertia of continuous body image32 A Hollow Sphere

About its diameter

moment of inertia of continuous body image33
moment of inertia of continuous body image34
Annular Disc

About an axis perpendicular to plane of disc

moment of inertia of continuous body image35
moment of inertia of continuous body image36 Thick Hollow cylinder

About axis of the cylinder

moment of inertia of continuous body image37
moment of inertia of continuous body image38
A Rectangular Plate

Axes lying in the plane of the plate and passing through c.m.

moment of inertia of continuous body image39

> moment of inertia of continuous body image40 , moment of inertia of continuous body image41
Axis perpendicular to plane and passing through its c.m.

moment of inertia of continuous body image42

moment of inertia of continuous body image43

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