Moment of inertia

Storyboard

The moment of inertia is the rotating factor that is equivalent to the mass in the translation.

The moment of inertia can be determined empirically by rotating a body around an axis or calculating how the mass is distributed around the axis.

>Model

ID:(600, 0)



Moment of inertia of a particle off axis

Equation

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A simple application of the Steiner's theorem is a point mass $m$ at a distance $L$ from an axis. Since a point mass has no dimensions, it has no moment of inertia with respect to its center of mass. However, since the center of mass is at a distance of $L$ from the axis according to Steiner's theorem,

$ I = I_{CM} + m d ^2$



its moment of inertia will be

$ I = m L ^2$

$I$
Moment of inertia of a mathematical pendulum
$kg m^2$
9799
$L$
Pendulum Length
$m$
6282
$m$
Point Mass
$kg$
6281

.

ID:(9880, 0)



Moment of inertia calculation method

Equation

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The total moment of inertia $I_t$ of an object is calculated by summing the moments of inertia of its components that are comparable to the moment of inertia of an individual particle,

$ I = m r ^2$



resulting in a moment of inertia as

$I_t=\sum_kI_k$

$I_k$
Moment of Inertia of k-th Element
$kg m^2$
6160
$I_t$
Moment of Inertia Total
$kg m^2$
6161

.

ID:(4438, 0)



Calculo del momento de inercia de un cuerpo

Equation

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Si el eje no varia y la distribución de la masa no varia el momento de inercia es constante se tiene que es

$ I =\displaystyle\int_V r ^2 \rho dV $

$I$
Constant moment of inertia
$kg m^2$
8765
$I_0$
Initial moment of inertia
$kg m^2$
8766

Para calcular el momento de inercia de un cuerpo se debe considerar este desglosado en pequeños volúmenes que se suman para obtener el momento de inercia total

$I_t=\sum_kI_k$



Como los momentos de inercia de masas m a una distancia r del eje son

$ I = m r ^2$

\\n\\npor lo que si se define la masa como la densidad \rho por el volumen dV se tiene que el momento de inercia es\\n\\n

$I=\displaystyle\sum_k I_k =\displaystyle\sum_k m_k r_k^2 = \displaystyle\sum_k \rho r_k^2 dV \rightarrow \displaystyle\int_V \rho r^2 dV$



por lo que

$ I =\displaystyle\int_V r ^2 \rho dV $

ID:(10583, 0)



Bar that rotates around an axis $\perp$

Image

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A bar with mass $m$ and length $l$ rotating around its center, which coincides with the center of mass:

ID:(10962, 0)



Moment of inertia of bar length $l$ axis $\perp$

Equation

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The moment of inertia of a rod rotating around a perpendicular ($\perp$) axis passing through the center is obtained by dividing the body into small volumes and summing them:

$ I =\displaystyle\int_V r ^2 \rho dV $



resulting in

$ I_{CM} =\displaystyle\frac{1}{12} m l ^2$

$m$
Body mass
$kg$
6150
$l$
Length of the Bar
$m$
6151
$I_{CM}$
Moment of Inertia at the CM of a thin Bar, perpendicular Axis
$kg m^2$
5323

.

ID:(4432, 0)



Cylinder rotating around axis $\parallel$

Image

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A rotation of a cylinder with mass $m$ and radius $r$ around the axis of the cylinder, where the center of mass (CM) is located at mid-height:

ID:(10964, 0)



Cylinder moment of inertia, axis $\parallel$

Equation

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The moment of inertia of a cylinder rotating around an axis parallel ($\parallel$) to its central axis is obtained by segmenting the body into small volumes and summing them:

$ I =\displaystyle\int_V r ^2 \rho dV $



resulting in

$ I_{CM} =\displaystyle\frac{1}{2} m r_c ^2$

$m$
Body mass
$kg$
6150
$I_{CM}$
Moment of Inertia at the CM of a Cylinder, Axis parallel to the Cylinder Axis
$kg m^2$
5324
$r_c$
Radius of a Cylinder
$m$
5319

.

ID:(4434, 0)



Cylinder that rotates about axis $\perp$

Image

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In this scenario, a cylinder with mass $m$, radius $r$, and height $h$ is rotating around an axis perpendicular to its own axis. This axis passes through the midpoint of the cylinder's length, where the center of mass (CM) is located:

ID:(10965, 0)



Cylinder moment of inertia, axis $\perp$

Equation

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The moment of inertia of a cylinder rotating around a perpendicular ($\perp$) axis passing through the center is obtained by segmenting the body into small volumes and summing them:

$ I =\displaystyle\int_V r ^2 \rho dV $



resulting in

$ I_{CM} =\displaystyle\frac{1}{12} m ( h ^2+3 r_c ^2)$

$m$
Body mass
$kg$
6150
$h$
Cylinder Height
$m$
5318
$I_{CM}$
Moment of Inertia at the CM of a Cylinder, Axis perpendicular to the Cylinder Axis
$kg m^2$
5325
$r_c$
Radius of a Cylinder
$m$
5319

.

ID:(4435, 0)



Regular parallelepiped moment of inertia

Image

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A straight rectangular parallelepiped with mass $m$ and sides $a$ and $b$, perpendicular to the axis of rotation, is rotating around its center of mass, which is located at the geometric center of the body:

ID:(10973, 0)



Moment of inertia of a right parallelepiped

Equation

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The moment of inertia of a parallelepiped rotating around an axis passing through its center is obtained by partitioning the body into small volumes and summing them up:

$ I =\displaystyle\int_V r ^2 \rho dV $



resulting in

$ I_{CM} =\displaystyle\frac{1}{12} m ( a ^2+ b ^2)$

$m$
Body mass
$kg$
6150
$a$
Length of the Edge of the Straight Parallelepiped
$m$
6152
$I_{CM}$
Moment of Inertia at the CM of Parallelepiped, Center to the Face
$kg m^2$
5322
$b$
Width of the Edge of the Straight Parallelepiped
$m$
6153

.

ID:(4433, 0)



Straight parallelepiped

Image

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In the case of a right rectangular parallelepiped with mass $m$ and side $a$, the center of mass is located at the geometric center:

ID:(10963, 0)



Momento de inercia de cubo recto

Equation

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El momento de inercia de un cubo que rota en torno a un eje que pasa por el centro se obtiene segmentando el cuerpo en pequeños volúmenes sumando:

$ I =\displaystyle\int_V r ^2 \rho dV $



resultando

$ I_{CM} =\displaystyle\frac{1}{6} m a ^2$

$m$
Body mass
$kg$
6150
$a$
Length of the Edge of the Straight Parallelepiped
$m$
6152
$I_{CM}$
Moment of Inertia at the CM of Parallelepiped, Center to the Face
$kg m^2$
5322

ID:(10972, 0)



Sphere

Image

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A sphere with mass $m$ and radius $r$ is rotating around its center of mass, which is located at its geometric center:

ID:(10490, 0)



Moment of inertia of a sphere

Equation

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The moment of inertia of a sphere rotating around an axis passing through its center is obtained by segmenting the body into small volumes and summing:

$ I =\displaystyle\int_V r ^2 \rho dV $



resulting in

$ I_{CM} =\displaystyle\frac{2}{5} m r_e ^2$

$m$
Body mass
$kg$
6150
$I_{CM}$
Moment of Inertia at the CM of a Sphere
$kg m^2$
5326
$r_e$
Radio of the Sphere
$m$
5321

.

ID:(4436, 0)



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