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Moment of Inertia

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 Sub Topics 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 asMoment of inertia = mass $\times$ radius2 (kgm2)IA = m $\times$ r2If there are large no of radius taken thenMoment of inertia = m1r12 + m2r22 + m3r32 + ..... + mnrn2Where n is the number of samples takenIA = $\sum$ mn rn2There are two factors that determine the moment of inertia of rotating bodyMass of the bodyRadius of the bodySince mass is expressed in kilogram and radius in meters.The unit of Moment of inertia is kilogram meter square (kgm2).

Moment of Inertia Definition

It is defined as the sum of the products of the mass and the square of the perpendicular distance to the axis of rotation of each particle in a body rotating about an axis. It is given as
I = Mr2Where M = mass of the body,
r = radius of body from axis of rotation.

Moment of  Inertia  of a Rectangle

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

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

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

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

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

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

Moment of Inertia of a Sphere

The moment of inertia of a solid sphere with the axis of rotation at center is
I = $\frac{2}{5}$ MR2Here M is the mass and R is the radius of axis of rotation.

Moment of Inertia of a Triangle

The moment of inertia of the triangle is
Ix = $\frac{1}{12}$ y3 dx
Ix = $\int$ 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 moment of inertia of triangle $\bar{I_x}$ = $\frac{1}{36}$ bh3 and Ix = $\frac{1}{12}$ bh3.

Moment of Inertia of Cylinder

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

Moment  of Inertia of a Circle

The moment of inertia of circle is
$\bar{I_x}$ = Iy = $\frac{1}{4}$ $\pi$ r4 and Jo = $\frac{1}{2}$ $\pi$ r4

Moment of Inertia of a Rod

The moment of inertia of rod is given by
I = $\int_0^M$ r2 dm = $\int_{-L/2}^{L/2}$ r2$\frac{M}{L}$ dr = $\frac{1}{12}$ ML2

Moment of Inertia of a Disk

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

Moment of Inertia of a Beam

The Moment of inertia of beam varies based on the inclination as shown in figure:

First Moment of Inertia

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,
Ix = $\int$ y dA Similarly, the first moment of inertia about y-axis would be given by,
Iy = $\int$ x dA

Second Moment of Inertia

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,
Ix = $\int y^{2} dA$
Similarly, the second moment of inertia about y-axis is given as
Iy = $\int x^{2} dA$
In the similar way you can calculate the third moment of inertia.

Polar Moment of Inertia

The polar moment of inertia is given by
J = $\sum$ x r2where r = the radius of small area, da from the perpendicular axis - for a plane area the perpendicular axis is a point

The polar moment of inertia is the sum of any two moments of inertia about axes which are at right angles to each other
J = Ixx + Iyywhere Ixx gives the moment of Inertia in x-axis
Iyy gives the moment of inertia in y-axis
or
The polar moment of inertia J of an element about an axis perpendicular to its plane is nothing but the product of the area of the element and square of its distance from the axis.

The polar moment of inertia is
dJ = $\rho$2 dA
= (x2+y2)dA
= dIy + dIx

Area Moment of Inertia

Lets see that in the figure the moment of area by summing together l2dA for all the given elements of area dA in the above region.

Hence Area moment of inertia
I = $\int$ A l2 dA For a rectangular region the area moment of inertia,
Ix = $\int$ A y2 dA
Iy = $\int$ A x2 dA

The above equations tells gives the area moment of inertia.

Mass moment of Inertia

The Mass Moment of Inertia of a solid tells the solid's ability to resist changes in rotational speed about a specific axis. The larger the Mass Moment of Inertia the smaller the angular acceleration would be about an axis for a given torque. The mass moment of inertia is the measurement of the distribution of the mass of an object or body relative to a given axis. It is represented by the symbol I.

For a particle of mass m the moment of inertia is
Io = MR2
Lets say that the body is made of sum of particles each having mass dm. The formula is
I = $\int \sum$ m r2 dm Similarly
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