D5. Mass distribution - Revision history

Eantem: /* D5.4 Steiner’s Theorem */

2026-02-24T18:23:25Z

D5.4 Steiner’s Theorem

← Older revision		Revision as of 18:23, 24 February 2026
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	'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia <span><math>(\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0)</math></span> for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>		'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia <span><math>(\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0)</math></span> for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>

	~~<span>(<math>\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0</math>)</span>~~
	{\|		{\|
	\|		\|

Eantem: /* D5.4 Steiner’s Theorem */

2026-02-24T18:23:00Z

D5.4 Steiner’s Theorem

← Older revision		Revision as of 18:23, 24 February 2026
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	When calculating the inertia tensor of a rigid body, it is necessary to make a qualitative assessment before resorting to the <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.5 Change of vector basis\|'''Table''']]</span>, since it only contains minimal information (the expression for an entire tensor is never given). This section presents some general properties that facilitate this assessment.<br>		When calculating the inertia tensor of a rigid body, it is necessary to make a qualitative assessment before resorting to the <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.5 Change of vector basis\|'''Table''']]</span>, since it only contains minimal information (the expression for an entire tensor is never given). This section presents some general properties that facilitate this assessment.<br>

	'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia <span>(<math>\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0</math>)</span> for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>		'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia <span><math>(\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0)</math></span> for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>

			<span>(<math>\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0</math>)</span>
	{\|		{\|
	\|		\|

Eantem: /* D5.4 Steiner’s Theorem */

2026-02-24T18:21:49Z

D5.4 Steiner’s Theorem

← Older revision		Revision as of 18:21, 24 February 2026
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	When calculating the inertia tensor of a rigid body, it is necessary to make a qualitative assessment before resorting to the <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.5 Change of vector basis\|'''Table''']]</span>, since it only contains minimal information (the expression for an entire tensor is never given). This section presents some general properties that facilitate this assessment.<br>		When calculating the inertia tensor of a rigid body, it is necessary to make a qualitative assessment before resorting to the <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.5 Change of vector basis\|'''Table''']]</span>, since it only contains minimal information (the expression for an entire tensor is never given). This section presents some general properties that facilitate this assessment.<br>

	'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia <span> (<math>\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0</math>) </span> for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>		'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia <span>(<math>\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0</math>)</span> for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>

	{\|		{\|

Eantem: /* D5.4 Steiner’s Theorem */

2026-02-24T18:20:50Z

D5.4 Steiner’s Theorem

← Older revision		Revision as of 18:20, 24 February 2026
Line 361:		Line 361:
	When calculating the inertia tensor of a rigid body, it is necessary to make a qualitative assessment before resorting to the <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.5 Change of vector basis\|'''Table''']]</span>, since it only contains minimal information (the expression for an entire tensor is never given). This section presents some general properties that facilitate this assessment.<br>		When calculating the inertia tensor of a rigid body, it is necessary to make a qualitative assessment before resorting to the <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.5 Change of vector basis\|'''Table''']]</span>, since it only contains minimal information (the expression for an entire tensor is never given). This section presents some general properties that facilitate this assessment.<br>

	'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia (<math>\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0</math>) for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>		'''Property 1:''' In a planar rigid body ('''Figure D5.6'''), the direction perpendicular to it (direction k) is always a principal direction of inertia <span> (<math>\mathrm{I}_\mathrm{ik}=\mathrm{I}_\mathrm{jk}=0</math>) </span> for any point <math>\Qs</math>, and the corresponding inertia moment is the sum of the other two (Pythagoras theorem):<br>

	{\|		{\|

Eantem: /* ✏️ Example D5.7: symmetrical rotor */

2026-02-24T18:02:33Z

✏️ Example D5.7: symmetrical rotor

← Older revision		Revision as of 18:02, 24 February 2026
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	'''Property 5''': When a rigid body has three or more equal moments of inertia in the same plane ij for a point <math>\Os</math>, it is also a '''symmetrical rotor for''' <math>\Os</math>:<math>\Bigr[\mathrm{II}(\Os) \Bigr]_\mathrm{ijk}=\Bigr[\mathrm{II}(\Os) \Bigr]_\mathrm{i'j'k'}</math>. The proof is longer than that of property 4, and is omitted.<br>		'''Property 5''': When a rigid body has three or more equal moments of inertia in the same plane ij for a point <math>\Os</math>, it is also a '''symmetrical rotor for''' <math>\Os</math>:<math>\Bigr[\mathrm{II}(\Os) \Bigr]_\mathrm{ijk}=\Bigr[\mathrm{II}(\Os) \Bigr]_\mathrm{i'j'k}</math>. The proof is longer than that of property 4, and is omitted.<br>

	<div>		<div>

	====✏️ Example D5.8: symmetrical rotor ====		====✏️ Example D5.8: symmetrical rotor ====
	------		------

Bros: /* D5.3 Some relevant properties of the inertia tensor */

2024-12-03T17:23:40Z

D5.3 Some relevant properties of the inertia tensor

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	</small>		</small>
			</div>

	------------		------------

Bros: /* ✏️ Example D5.5: discrete rigid body */

2024-12-03T17:23:08Z

✏️ Example D5.5: discrete rigid body

← Older revision		Revision as of 17:23, 3 December 2024
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	</small>		</small>
	~~</div>~~

	------------		------------

Bros at 00:12, 3 December 2024

2024-12-03T00:12:39Z

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	<math>\overline{\Os_\Rs\Gs}=\frac{1}{\mathrm{M}} \int_\mathrm{S}\mathrm{dm}(\Ps)\overline{\Os_\Rs\Ps}</math><br>		<math>\overline{\Os_\Rs\Gs}=\frac{1}{\mathrm{M}} \int_\mathrm{S}\mathrm{dm}(\Ps)\overline{\Os_\Rs\Ps}</math><br>

	The <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.~~5 Change~~ of ~~vector basis~~\|'''Table''']]</span> shows the centre of mass of the most common geometries. From this information and for rigid bodies S formed by several of these elements <math>\mathrm{S}_\is</math>, the position of the centre of mass can be found as a weighted average of the position of each <math>\mathrm{G}_\is</math>.		The <span style="text-decoration: underline;"> [[D5. Mass distribution#D5.7 Table of centers of mass and inertia tensors of homogeneous rigid bodies\|'''Table''']]</span> shows the centre of mass of the most common geometries. From this information and for rigid bodies S formed by several of these elements <math>\mathrm{S}_\is</math>, the position of the centre of mass can be found as a weighted average of the position of each <math>\mathrm{G}_\is</math>.

	<div>		<div>

Bros at 00:12, 3 December 2024

2024-12-03T00:12:16Z

← Older revision		Revision as of 00:12, 3 December 2024
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	------------		------------
	------------		------------
	==D5.1 ~~Centre~~ of masses==		==D5.1 Center of masses==

	The <span style="text-decoration: underline;"> [[D4. Vectorial theorems#D4.1 Teorema de la Quantitat de Moviment (TQM) en referències galileanes\|'''centre of mass''']]</span> of a mechanical system is a point whose kinematics is a weighted kinematics of all the elements of the system that have mass. In this course it is represented by the letter '''G'''.		The <span style="text-decoration: underline;"> [[D4. Vectorial theorems#D4.1 Teorema de la Quantitat de Moviment (TQM) en referències galileanes\|'''centre of mass''']]</span> of a mechanical system is a point whose kinematics is a weighted kinematics of all the elements of the system that have mass. In this course it is represented by the letter '''G'''.

Bros at 17:41, 30 November 2024

2024-11-30T17:41:10Z

← Older revision		Revision as of 17:41, 30 November 2024
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	~~<div>~~
	<div>		<div>
	====✏️ Example D5.5: discrete rigid body ====		====✏️ Example D5.5: discrete rigid body ====