Moment of inertia - Revision history

Vipul: /* For typical shapes */

2010-09-16T23:22:52Z

For typical shapes

Vipul: /* For typical shapes */

2010-01-20T22:41:58Z

For typical shapes

← Older revision		Revision as of 22:41, 20 January 2010
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	\| closed hollow hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(19/18)mr^2</math> \|\| <math>r\sqrt{19/18}</math>		\| closed hollow hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(19/18)mr^2</math> \|\| <math>r\sqrt{19/18}</math>
	\|-		\|-
	\| solid right circular cone of base radius <math>r</math>, height <math>h</math> \|\| axis through the vertex, along height, perpendicular to base \|\| <math>(3/10)mr^2</math> \|\| <math>r\sqrt{3/10}		\| solid right circular cone of base radius <math>r</math>, height <math>h</math> \|\| axis through the vertex, along height, perpendicular to base \|\| <math>(3/10)mr^2</math> \|\| <math>r\sqrt{3/10}</math>
	\|-		\|-
	\| thin solid circular disk of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>		\| thin solid circular disk of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>

Vipul: /* For typical shapes */

2010-01-20T22:41:29Z

For typical shapes

← Older revision		Revision as of 22:41, 20 January 2010
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Shape and parameters !! Axis !! Moment of inertia		! Shape and parameters !! Axis !! Moment of inertia !! Radius of gyration
	\|-		\|-
	\| solid cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>(1/2)mr^2</math>		\| solid cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>
	\|-		\|-
	\| open-ended hollow cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>mr^2</math>		\| open-ended hollow cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>mr^2</math> \|\| <math>r</math>
	\|-		\|-
	\| solid sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/5)mr^2</math>		\| solid sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/5)mr^2</math> \|\| <math>r\sqrt{2/5}</math>
	\|-		\|-
	\| hollow sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/3)mr^2</math>		\| hollow sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/3)mr^2</math> \|\| <math>r\sqrt{2/3}</math>
	\|-		\|-
	\| solid hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(2/5)mr^2</math>		\| solid hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(2/5)mr^2</math> \|\| <math>r\sqrt{2/5}</math>
	\|-		\|-
	\| closed hollow hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(19/18)mr^2</math> (?)		\| closed hollow hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(19/18)mr^2</math> \|\| <math>r\sqrt{19/18}</math>
			\|-
			\| solid right circular cone of base radius <math>r</math>, height <math>h</math> \|\| axis through the vertex, along height, perpendicular to base \|\| <math>(3/10)mr^2</math> \|\| <math>r\sqrt{3/10}
			\|-
			\| thin solid circular disk of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>
			\|-
			\| thin solid circular disk of radius <math>r</math> \|\| axis through center, a diameter of circle \|\| <math>(1/4)mr^2</math> \|\| <math>r/2</math>
			\|-
			\| circular hoop (thin hollow circle) of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>mr^2</math> \|\| <math>r</math>
			\|-
			\| circular hoop (thin hollow circle) of radius <math>r</math> \|\| axis through center, a diameter of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>
	\|}		\|}

Vipul at 19:31, 20 January 2010

2010-01-20T19:31:49Z

← Older revision		Revision as of 19:31, 20 January 2010
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			{{angular analogue of\|mass}}
	==Definition==		==Definition==

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	==Units and dimensions==		==Units and dimensions==

	~~The~~ MLT dimensions ~~of moment of inertia are~~ <math>ML^2</math>~~, and the~~ SI units ~~are~~ <math>kgm^2</math> (kilograms meter-squared).		{\| class="sortable" border="1"
			! Question !! Answer
			\|-
			\| Scalar or vector? \|\| [[Quantity type::Scalar]]
			\|-
			\| MLT dimensions \|\| <math>ML^2</math>: [[MLT::MLT;1;2;0]]
			\|-
			\| SI units \|\| <math>kgm^2</math> (kilograms meter-squared)
			\|}
	==Main facts==		==Main facts==

Vipul: Created page with '==Definition== ===For a single particle about an axis=== The moment of inertia of a particle of mass $m$ about an axis $\ell$ is defined as $mr^2$
2010-01-20T00:56:46Z

Created page with '==Definition== ===For a single particle about an axis=== The moment of inertia of a particle of mass <math>m</math> about an axis <math>\ell</math> is defined as <math>mr^2</ma…'

New page
==Definition==

===For a single particle about an axis===

The moment of inertia of a particle of mass <math>m</math> about an axis <math>\ell</math> is defined as <math>mr^2</math>, where <math>r</math> is the perpendicular distance from <math>m</math> to <math>\ell</math>.

===For a finite collection of particles about an axis===

The moment of inertia about a system of particles <math>m_1, m_2, \dots, m_n</math> about an axis <math>\ell</math> is defined as:

<math>\sum_{i=1}^n m_ir_i^2 = m_1r_1^2 + m_2r_2^2 + \dots + m_nr_n^2</math>

where <math>r_i</math> is the perpendicular distance from <math>m_i</math> to <math>\ell</math>.

===For a rigid body about an axis===

The moment of inertia of a rigid body about an axis <math>\ell</math> is defined as:

<math>\int r^2 \, dm</math>

where, for a mass differential <math>dm</math> of the body, <math>r</math> is defined as the perpendicular distance from the mass differential to <math>\ell</math>. This is equivalent to:

<math>\int r^2 \rho \, dV</math>

where <math>dV</math> is the volume differential and <math>\rho</math> is the [[density]]. For a body of constant density, <math>\rho</math> can be pulled out of the integral, and we obtain:

<math>\rho \int r^2 \, dV</math>

==Units and dimensions==

The MLT dimensions of moment of inertia are <math>ML^2</math>, and the SI units are <math>kgm^2</math> (kilograms meter-squared).

==Main facts==

There are two main facts used to compute moment of inertia:

* [[Parallel axis theorem]]
* [[Perpendicular axis theorem]]

==For typical shapes==

Below is the moment of inertia of bodies of constant [[density]] and mass <math>m</math> with some typical shapes. Note that for the first four of these shapes, all rotations about the specified axis are symmetries of the figure, so performing these rotations does not change the geometry in so far as contact with other surfaces is concerned:

{| class="sortable" border="1"
! Shape and parameters !! Axis !! Moment of inertia
|-
| solid cylinder of base radius <math>r</math> || axis of cylinder, passes through centers of all the circles || <math>(1/2)mr^2</math>
|-
| open-ended hollow cylinder of base radius <math>r</math> || axis of cylinder, passes through centers of all the circles || <math>mr^2</math>
|-
| solid sphere of radius <math>r</math> || any axis through the center of the sphere || <math>(2/5)mr^2</math>
|-
| hollow sphere of radius <math>r</math> || any axis through the center of the sphere || <math>(2/3)mr^2</math>
|-
| solid hemisphere of radius <math>r</math> || axis through the center perpendicular to the bounding circular disk || <math>(2/5)mr^2</math>
|-
| closed hollow hemisphere of radius <math>r</math> || axis through the center perpendicular to the bounding circular disk || <math>(19/18)mr^2</math> (?)
|}

← Older revision		Revision as of 23:22, 16 September 2010
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Shape and parameters !! Axis !! Moment of inertia !! Radius of gyration		! Shape and parameters !! Axis !! Moment of inertia !! Radius of gyration !! Quotient <math>I/mr^2</math> where <math>r</math> is a natural radius of rotation about that axis
	\|-		\|-
	\| solid cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>		\| solid cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math> \|\| 1/2
	\|-		\|-
	\| open-ended hollow cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>mr^2</math> \|\| <math>r</math>		\| open-ended hollow cylinder of base radius <math>r</math> \|\| axis of cylinder, passes through centers of all the circles \|\| <math>mr^2</math> \|\| <math>r</math> \|\| 1
	\|-		\|-
	\| solid sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/5)mr^2</math> \|\| <math>r\sqrt{2/5}</math>		\| solid sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/5)mr^2</math> \|\| <math>r\sqrt{2/5}</math> \|\| 2/5
	\|-		\|-
	\| hollow sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/3)mr^2</math> \|\| <math>r\sqrt{2/3}</math>		\| hollow sphere of radius <math>r</math> \|\| any axis through the center of the sphere \|\| <math>(2/3)mr^2</math> \|\| <math>r\sqrt{2/3}</math> \|\| 2/3
	\|-		\|-
	\| solid hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(2/5)mr^2</math> \|\| <math>r\sqrt{2/5}</math>		\| solid hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(2/5)mr^2</math> \|\| <math>r\sqrt{2/5}</math> \|\| --
	\|-		\|-
	\| closed hollow hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(19/18)mr^2</math> \|\| <math>r\sqrt{19/18}</math>		\| closed hollow hemisphere of radius <math>r</math> \|\| axis through the center perpendicular to the bounding circular disk \|\| <math>(19/18)mr^2</math> \|\| <math>r\sqrt{19/18}</math> \|\| --
	\|-		\|-
	\| solid right circular cone of base radius <math>r</math>, height <math>h</math> \|\| axis through the vertex, along height, perpendicular to base \|\| <math>(3/10)mr^2</math> \|\| <math>r\sqrt{3/10}</math>		\| solid right circular cone of base radius <math>r</math>, height <math>h</math> \|\| axis through the vertex, along height, perpendicular to base \|\| <math>(3/10)mr^2</math> \|\| <math>r\sqrt{3/10}</math> \|\| --
	\|-		\|-
	\| thin solid circular disk of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>		\| thin solid circular disk of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math> \|\| 1/2
	\|-		\|-
	\| thin solid circular disk of radius <math>r</math> \|\| axis through center, a diameter of circle \|\| <math>(1/4)mr^2</math> \|\| <math>r/2</math>		\| thin solid circular disk of radius <math>r</math> \|\| axis through center, a diameter of circle \|\| <math>(1/4)mr^2</math> \|\| <math>r/2</math> \|\| --
	\|-		\|-
	\| circular hoop (thin hollow circle) of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>mr^2</math> \|\| <math>r</math>		\| circular hoop (thin hollow circle) of radius <math>r</math> \|\| axis through center, perpendicular to plane of circle \|\| <math>mr^2</math> \|\| <math>r</math> \|\| 1
	\|-		\|-
	\| circular hoop (thin hollow circle) of radius <math>r</math> \|\| axis through center, a diameter of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math>		\| circular hoop (thin hollow circle) of radius <math>r</math> \|\| axis through center, a diameter of circle \|\| <math>(1/2)mr^2</math> \|\| <math>r/\sqrt{2}</math> \|\| --
	\|}		\|}