Moment of Inertia

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Sub Topics Moment of Inertia Definition Moment of  Inertia  of a Rectangle Moment of Inertia of a Sphere Moment of Inertia of a Triangle Moment of Inertia of Cylinder Moment  of Inertia of a Circle Moment of Inertia of a Rod Moment of Inertia of a Disk Moment of Inertia of a Beam First Moment of Inertia Second Moment of Inertia Polar Moment of Inertia Area Moment of Inertia Mass moment of Inertia Moment of Inertia Table Rotational Moment of Inertia Calculate Moment of Inertia

The inertia is that where a body offers a resistance to move linearly and is directly proportional to the mass. If this reluctance is seen in a rotating body about an axis we use the term moment of inertia. It gives the distribution of mass about the axis of rotation. It is given as Moment of inertia = mass $\times$ radius2 (kgm2) IA = m $\times$ r2 If there are large no of radius taken then Moment of inertia = m1r12 + m2r22 + m3r32 + ..... + mnrn2 Where n is the number of samples taken IA = $\sum$ mn rn2 There are two factors that determine the moment of inertia of rotating body Mass of the body Radius of the body Since mass is expressed in kilogram and radius in meters.The unit of Moment of inertia is kilogram meter square (kgm2).

The Moment of inertia of rectangle varies in x and y axes as shown in figure

The moment of inertia I_x = $\int_A$ y² dA
= $\int_0^h$ y² (b dy)
= $\frac{hy^3}{3}$ |₀^h
= $\frac{bh^3}{3}$

I_x = I_x'= $\int_A$ y² dA
= 4 $\int_0^h$ y2 ($\frac{b}{2}$ dy)
= 4 $\frac{b}{2}$ $\frac{y^3}{3}$|₀^h/2
= $\frac{bh^3}{12}$

The moment of inertia I_y = $\int_A$ x² dA
= $\int_0^b$ x² (h dx)
= $\frac{hx^3}{3}$|₀^b
= $\frac{hb^3}{3}$

Iy = Iy'= $\int_A$ x² dA
= 4 $\int_0^h$ x² ($\frac{h}{2}$ dx)
= 4 $\frac{h}{2}$ $\frac{x^3}{3}$|₀^b/2
= $\frac{hb^3}{12}$

I_x = $\bar{I_x}$ + Ad²
= $\frac{bh^3}{12}$ + (bh) $\frac{h}{2}^2$
= $\frac{bh^3}{12}$ + $\frac{bh^3}{4}$
= $\frac{bh^3}{3}$.

The moment of inertia of a solid sphere with the axis of rotation at center is

The moment of inertia of the triangle is

I_x = $\frac{1}{12}$ y³ dx

I_x = $\int$ y² dM = $\int$ $\int$ y² $\rho$ dy dx

Here $\rho$ is density to integrate over volume then y goes from 0 to y = h - x and x goes from 0 to b.

The Moments of Inertia of cylinder is

• Solid cylinder with axis of rotation at the center = $\frac{1}{2}$ M R²
• Solid cylinder with axis of rotation as surface = $\frac{3}{2}$ M R²
• Hallow cylinder with axis of rotation as center = M R²
• Hallow cylinder with axis of rotation as surface = 2 MR²

The moment of inertia of circle is
$\bar{I_x}$ = I_y = $\frac{1}{4}$ $\pi$ r⁴ and J_o = $\frac{1}{2}$ $\pi$ r⁴
The moment of inertia of rod is given by

The moment of inertia along radial axis is given by I = $\frac{1}{2}$ MR²
The moment of inertia along the central axis is given by I = $\frac{1}{4}$ MR²
The Moment of inertia of beam varies based on the inclination as shown in figure:
The first moment of inertia tells that the moment arm will be raised to a power of one. The first moment of any area calculated about x-axis would be given by,

I_y = $\int$ x dA

The second moment of inertia is that the moment arm is raised to a power of 2. So the second moment about x-axis is given by,

I_x = $\int y^{2} dA $

Similarly, the second moment of inertia about y-axis is given as

I_y = $\int x^{2} dA $

The polar moment of inertia is the sum of any two moments of inertia about axes which are at right angles to each other

The polar moment of inertia is

dJ = $\rho$² dA
= (x²+y²)dA
= dI_y + dI_x

Hence Area moment of inertia

I_x = $\int$ A y² dA
I_y = $\int$ A x² dA

The above equations tells gives the area moment of inertia.

For a particle of mass m the moment of inertia is

I = $\int \sum$ r² p $\rho$ V

where $\rho$ = density and for area dA the moment of inertia is

I = $\int \sum$ r² p dA

Here is the table for moment of inertia


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