Physics MOMENT OF INERTIA

Moment of Inertia :

Consider a rigid body rotating about a fixed axis `AB`. The particle `P` of mass `m_i` on the body describes a circle of radius `r_i`.

The tangential force on the particle `m_i` is `F_t=(mdv)/(dt) =m((dv)/(dt))`

As `v=omega r ,(dv)/(dt)=(r d omega)/(dt)`
`F_t mr (d omega)/(dt)=mr * a`

The radial force `F_r = mw^2r` . The torque of the radial force is zero, as it passes through the axis, whereas the torque of the tangential force

`t=(f_1) * (r)=(m ra)(r)=> t_i=(m_ir_i^2)a`

Summing over, all the particles, the total torque of all the forces acting on all the particles of the body is,

`t_(text(total))=St_i=[Sigma m_i r_i^2] * a`

`t_(text(total))=Ia`

where `I=Sigma m_ir_i^2I` is called moment of inertia

Moment of Inertia of Continuous Bodies :

For a system of discrete particles, moment of inertia `I=Sigma m_ir_i^2` but for a system of continuous mass

(i) `I=int r^2 dm`

We illustrate for some bodies the above process

Moment of inertia of a rod for an axis passing through its `CM`.

Let us assume a rod of mass `M` and length `L` as shown. Consider an element `dx` at a distance `x` from the axis. the mass of the element

`dm=(M/L)*dx`

`dI=m^2dm=x^2(M/L)*dx`

`I=M/L int_(-L/2)^(L/2) x^2dx`

The rod varies from `-L/2` on one side to `L/2` on the other side, hence the limits.

`=> I=M/L(x^2/3)_(-L/2)^(L/2)`

`= M/(3L) [(L/2)^2-(-L/2)^2]`

`= ML/2 [(L/2)^2-(L/2)^2]`

`=M/(3L)[L^2/8 xx 2]=(ML^2)/12`

Moment of Inertia of a Disc for an axis passing through its `CM` and normal to the plane of the disc.

Consider a disc of mass `M` and radius `R`. We choose an elemental ring of radius `r` and thickness `dr`, concentric with the disc. The lemental mass of the ring is `dm =(M/(pi R^2))(2prdr)`

`dI=r^2 dm=r^2(M/(pi R^2))(2prdr)`

`dI=r^2dm=r^2(M/(pi R^2))(2prdr)`

`dI=(2M)/R^2 r^3 dr`

`I=(2M)/(R^2) int_0^pi r^3 dr`

The disc can be thought of as concentric rings set from `r` equal to `0` to `R`.

`=> I=(2M)/R^2(r^4/4)^R=(2M)/R^2 * R^4/4=(MR^2)/2`

By choosing suitable elements, moments of inertia for other continuous bodies can be derived.