D1. Foundational laws of Newtonian dynamics - Revision history

Bros: /* ✏️ Example D1-7.1 */

2025-05-08T12:17:28Z

✏️ Example D1-7.1

← Older revision		Revision as of 12:17, 8 May 2025
Line 308:		Line 308:
	<math>\Fcal{-Cor}{Plat\rightarrow\Qs}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})</math><br>		<math>\Fcal{-Cor}{Plat\rightarrow\Qs}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})</math><br>

	As the transportation acceleration is orthogonal to <math>\OQvec</math> and <math>\overline{\boldsymbol{\Omega}}_0</math> is vertical, the Coriolis force is radial centripetal (towards '''O'''), with value <math>\ms\Rs\Omega_0^2</math>.		As the transportation acceleration is orthogonal to <math>\OQvec</math> and <math>\overline{\boldsymbol{\Omega}}_0</math> is vertical, the Coriolis force is radial centripetal (towards '''O'''), with value <math>2\ms\Rs\Omega_0^2</math>.

	<center>[[File:ExD1-7-2-eng2.png\|thumb\|center\|300px\|link=]]</center>		<center>[[File:ExD1-7-2-eng2.png\|thumb\|center\|300px\|link=]]</center>

Bros: /* ✏️ Example D1-7.1 */

2024-11-30T17:23:49Z

✏️ Example D1-7.1

← Older revision		Revision as of 17:23, 30 November 2024
Line 310:		Line 310:
	As the transportation acceleration is orthogonal to <math>\OQvec</math> and <math>\overline{\boldsymbol{\Omega}}_0</math> is vertical, the Coriolis force is radial centripetal (towards '''O'''), with value <math>\ms\Rs\Omega_0^2</math>.		As the transportation acceleration is orthogonal to <math>\OQvec</math> and <math>\overline{\boldsymbol{\Omega}}_0</math> is vertical, the Coriolis force is radial centripetal (towards '''O'''), with value <math>\ms\Rs\Omega_0^2</math>.

			<center>[[File:ExD1-7-2-eng2.png\|thumb\|center\|300px\|link=]]</center>
	<center>[[File:ExD1-7-2-~~eng~~.png\|thumb\|center\|300px\|link=]]</center>
	</small>		</small>

Bros: /* D1.5 Newton’s second law (fundamental law of dynamics) */

2024-11-30T17:21:32Z

D1.5 Newton’s second law (fundamental law of dynamics)

← Older revision		Revision as of 17:21, 30 November 2024
Line 212:		Line 212:

	:* Consequently, the <math>\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}</math> force that '''Q''' exerts on '''P''' will also have the <math>\QPvec</math> direction.		:* Consequently, the <math>\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}</math> force that '''Q''' exerts on '''P''' will also have the <math>\QPvec</math> direction.
	:* If we eliminate the '''Q''' particle, the parameter <math>\ms_\Ps</math> becomes irrelevant: <math>\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\~~ovtheerline~~{0}</math> regardless of its mass. Mass, then, only manifests itself in interaction.		:* If we eliminate the '''Q''' particle, the parameter <math>\ms_\Ps</math> becomes irrelevant: <math>\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\overline{0}</math> regardless of its mass. Mass, then, only manifests itself in interaction.
	the		the
	Let us now go back to the general case of many particles interacting with '''P'''. Newton's second law contains a '''principle of superposition''': the resultant force acting on '''P''' is the sum of those that each of the other particles would separately exert on '''P''': the simultaneous existence of several particles does not alter the interactions between them ('''Figure D1.7''').		Let us now go back to the general case of many particles interacting with '''P'''. Newton's second law contains a '''principle of superposition''': the resultant force acting on '''P''' is the sum of those that each of the other particles would separately exert on '''P''': the simultaneous existence of several particles does not alter the interactions between them ('''Figure D1.7''').

Bros at 17:20, 30 November 2024

2024-11-30T17:20:45Z

← Older revision		Revision as of 17:20, 30 November 2024
Line 282:		Line 282:
	The inertia force may be decomposed into two forces: the '''inertia transportation force''' and the '''Coriolis inertia force''':		The inertia force may be decomposed into two forces: the '''inertia transportation force''' and the '''Coriolis inertia force''':
	<center>		<center>
	<math>\Fcal{-}{NGal\rightarrow\Ps}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \Fcal{~~Cor~~}{NGal\rightarrow\Ps~~}^{-tr~~}+\Fcal{Cor}{NGal\rightarrow\Ps~~}^{-Cor~~} </math>.		<math>\Fcal{-}{NGal\rightarrow\Ps}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \Fcal{-tr}{NGal\rightarrow\Ps}+\Fcal{-Cor}{NGal\rightarrow\Ps} </math>.
	</center>		</center>

Bros at 17:17, 30 November 2024

2024-11-30T17:17:46Z

← Older revision		Revision as of 17:17, 30 November 2024
Line 46:		Line 46:
	\newcommand{\abs}[1]{\left\|{#1}\right\|}		\newcommand{\abs}[1]{\left\|{#1}\right\|}
	\newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}		\newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}}
			\newcommand{\Fcal}[2]{\vec{\cal{F}}^{\text{#1}}_{#2}}
	\newcommand{\vector}[3]{		\newcommand{\vector}[3]{
	\begin{Bmatrix}		\begin{Bmatrix}
Line 211:		Line 212:

	:* Consequently, the <math>\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}</math> force that '''Q''' exerts on '''P''' will also have the <math>\QPvec</math> direction.		:* Consequently, the <math>\overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}</math> force that '''Q''' exerts on '''P''' will also have the <math>\QPvec</math> direction.
	:* If we eliminate the '''Q''' particle, the parameter <math>\ms_\Ps</math> becomes irrelevant: <math>\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\~~overline~~{0}</math> regardless of its mass. Mass, then, only manifests itself in interaction.		:* If we eliminate the '''Q''' particle, the parameter <math>\ms_\Ps</math> becomes irrelevant: <math>\overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})=\ovtheerline{0}</math> regardless of its mass. Mass, then, only manifests itself in interaction.
			the
	Let us now go back to the general case of many particles interacting with '''P'''. Newton's second law contains a '''principle of superposition''': the resultant force acting on '''P''' is the sum of those that each of the other particles would separately exert on '''P''': the simultaneous existence of several particles does not alter the interactions between them ('''Figure D1.7''').		Let us now go back to the general case of many particles interacting with '''P'''. Newton's second law contains a '''principle of superposition''': the resultant force acting on '''P''' is the sum of those that each of the other particles would separately exert on '''P''': the simultaneous existence of several particles does not alter the interactions between them ('''Figure D1.7''').

Line 261:		Line 262:
	<center>		<center>
	Dynamics in Galilean reference frames: <math>\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})</math><br>		Dynamics in Galilean reference frames: <math>\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})</math><br>
	Dynamics in non Galilean reference frames: <math>\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\~~mathbf~~{F}_{\rightarrow \~~mathrm{P}}^{-~~}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})</math><br>		Dynamics in non Galilean reference frames: <math>\sum_{\mathrm{Q}} \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}}+\Fcal{-}{NGal\rightarrow\Ps}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})</math><br>
	</center>		</center>

	Subtracting the first equation from the second:		Subtracting the first equation from the second:
	<center>		<center>
	<math>\~~mathbf~~{F}_{\rightarrow \~~mathrm{P}}^{-~~}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})</math>		<math>\Fcal{-}{NGal\rightarrow\Ps}=\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{NGal}}(\mathbf{P})-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{Gal}}(\mathbf{P})</math>
	</center>		</center>

Line 281:		Line 282:
	The inertia force may be decomposed into two forces: the '''inertia transportation force''' and the '''Coriolis inertia force''':		The inertia force may be decomposed into two forces: the '''inertia transportation force''' and the '''Coriolis inertia force''':
	<center>		<center>
	<math>\~~mathbf~~{F}_{\rightarrow \~~mathrm{P}}^{-~~}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \~~mathbf~~{F}_{~~\mathrm{ar}~~ \rightarrow \~~mathrm{P}~~}^{-}+\~~mathbf~~{F}_{~~\text {cor }~~ \rightarrow \~~mathrm{P}~~}^{-} </math>.		<math>\Fcal{-}{NGal\rightarrow\Ps}=\left[-\mathrm{m}_{\mathrm{P}} \overline{\mathbf{a}}_{\mathrm{ar}}(\mathbf{P})\right]+\left[-\mathrm{m}_{\mathrm{P}} 2 \overline{\boldsymbol{\Omega}}_{\text {Gal }}^{\mathrm{NGal}} \times \overline{\mathbf{v}}_{\mathrm{NGal}}(\mathbf{P})\right] \equiv \Fcal{Cor}{NGal\rightarrow\Ps}^{-tr}+\Fcal{Cor}{NGal\rightarrow\Ps}^{-Cor} </math>.
	</center>		</center>

Line 302:		Line 303:
	::If '''P''' studies the <u>dynamics of '''Q''' from the rotating platform reference frame</u> (which has the centre '''O''' fixed to the ground and rotates with constant angular velocity <math>\Omega_0</math>), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:		::If '''P''' studies the <u>dynamics of '''Q''' from the rotating platform reference frame</u> (which has the centre '''O''' fixed to the ground and rotates with constant angular velocity <math>\Omega_0</math>), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:

	* Inertia transportation force <math>\~~mathbf~~{F}_{~~\mathrm{ar}~~\rightarrow \~~mathrm{Q}}^{-~~}</math>: as the Q <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''transportation motion''']]</span> is circular with centre '''O''', radius R (<math>=\mid \OQvec \mid</math>) and associated angular velocity <math>\Omega_0</math>, the transportation force will be radial and outwards (centrifugal) with value <math>\ms\Rs\Omega_0^2</math>.		* Inertia transportation force <math>\Fcal{-tr}{Plat\rightarrow\Qs}</math>: as the Q <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''transportation motion''']]</span> is circular with centre '''O''', radius R (<math>=\mid \OQvec \mid</math>) and associated angular velocity <math>\Omega_0</math>, the transportation force will be radial and outwards (centrifugal) with value <math>\ms\Rs\Omega_0^2</math>.

	* Coriolis inertia force : it is obtained from the <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''Coriolis acceleration''']]</span>:		* Coriolis inertia force <math>\Fcal{-Cor}{Plat\rightarrow\Qs}</math>: it is obtained from the <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''Coriolis acceleration''']]</span>:
	<math>\~~mathbf~~{F}~~_{\text~~ {~~Cor }~~ \rightarrow \~~mathbf{Q}}^{-~~}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})</math><br>		<math>\Fcal{-Cor}{Plat\rightarrow\Qs}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})</math><br>

	As the transportation acceleration is orthogonal to <math>\OQvec</math> and <math>\overline{\boldsymbol{\Omega}}_0</math> is vertical, the Coriolis force is radial centripetal (towards '''O'''), with value <math>\ms\Rs\Omega_0^2</math>.		As the transportation acceleration is orthogonal to <math>\OQvec</math> and <math>\overline{\boldsymbol{\Omega}}_0</math> is vertical, the Coriolis force is radial centripetal (towards '''O'''), with value <math>\ms\Rs\Omega_0^2</math>.

Bros: /* D1.1 Galilean reference frames */

2024-11-30T17:02:18Z

D1.1 Galilean reference frames

← Older revision		Revision as of 17:02, 30 November 2024
Line 112:		Line 112:
	</small></center>		</small></center>

	:* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, <math>\Ps</math> initially moves in the longitudinal direction of the frame forward <math>(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)</math>, and the springs deform with opposite signs (one stretches and the other shortens, '''Figure D1.3a'''); if it is oriented in the transverse direction of the wagon, <math>\Ps</math> initially moves in the transverse direction of the frame forward <math>(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)</math>, and the change of the springs length is the same ('''Figure D1.3b'''); if it is oriented in any other direction, the initial velocity of P has two components (<math>\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0</math>,<math>\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0</math>, '''Figure D1.3c'''). To summarize: the result of this experiment is <math>\vel{P,t}{R} \neq 0</math>, and it is independent from the position and the time instant, but not from the orientation. For this experiment, <u>the space of that reference frame is homogeneous but not isotropic, and time is uniform</u>.		:* If the surface is that of a wagon with constant braking acceleration relative to the ground (reference frame R2): if the frame is fixed to R2 in any position but oriented in the longitudinal direction of the wagon, <math>\Ps</math> initially moves in the longitudinal direction of the frame forward <math>(\left.\vel{P,t}{R}\right]_{\text{long frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {trans frame}}=0)</math>, and the springs deform with opposite signs (one stretches and the other shortens, '''Figure D1.3a'''); if it is oriented in the transverse direction of the wagon, <math>\Ps</math> initially moves in the transverse direction of the frame forward <math>(\left.\vel{P,t}{R}\right]_{\text{trans frame}} \neq 0,\left.\vel{P,t}{R}\right]_{\text {long frame}}=0)</math>, and the change of the springs length is the same ('''Figure D1.3b'''); if it is oriented in any other direction, the initial velocity of P has two components (<math>\left.\vel{P,t}{R}\right]_{\text{long marc }} \neq 0</math>,<math>\left.\vel{P,t}{R}\right]_{\text {trans marc }}\neq 0</math>, '''Figure D1.4c'''). To summarize: the result of this experiment is <math>\vel{P,t}{R} \neq 0</math>, and it is independent from the position and the time instant, but not from the orientation. For this experiment, <u>the space of that reference frame is homogeneous but not isotropic, and time is uniform</u>.

	If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.		If the wagon had variable acceleration, the time would not be uniform: when braking, the initial speed of P would be as discussed, but when accelerating, the initial speed would have a backward component instead of a forward component. Therefore, the initial speed would not be the same depending on the time instant.

Bros at 22:23, 5 November 2024

2024-11-05T22:23:16Z

← Older revision		Revision as of 22:23, 5 November 2024
Line 257:		Line 257:
	==D1.7 Particle dynamics in non Galilean reference frames==		==D1.7 Particle dynamics in non Galilean reference frames==

	As stated in <span style="text-decoration: underline;"> [[D1. ~~Lleis fundacionals de la mecànica newtoniana~~#D1.1 Galilean reference frames\|'''section D1.1''']]</span>, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:		As stated in <span style="text-decoration: underline;"> [[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames\|'''section D1.1''']]</span>, the dynamics in Galilean reference frames leads to the introduction of inertial forces on the particles that come from the characteristics of space-time, which is no longer neutral:

	<center>		<center>

Apons: /* ✏️ Example D1-7.1 */

2024-11-05T20:46:28Z

✏️ Example D1-7.1

← Older revision		Revision as of 20:46, 5 November 2024
Line 298:		Line 298:
	:: Two people, modeled as two particles '''P''' and '''Q''' with the same mass m, are at rest relative to a rotating platform and the ground, respectively.		:: Two people, modeled as two particles '''P''' and '''Q''' with the same mass m, are at rest relative to a rotating platform and the ground, respectively.

	:: If '''Q''' studies the <u>dynamics of '''P''' from the ground reference</u>, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.		:: If '''Q''' studies the <u>dynamics of '''P''' from the ground reference</u>, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.<br>

	::If '''P''' studies the <u>dynamics of '''Q''' from the rotating platform reference frame</u> (which has the centre '''O''' fixed to the ground and rotates with constant angular velocity <math>\Omega_0</math>), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:		::If '''P''' studies the <u>dynamics of '''Q''' from the rotating platform reference frame</u> (which has the centre '''O''' fixed to the ground and rotates with constant angular velocity <math>\Omega_0</math>), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:

Apons: /* ✏️ Example D1-7.1 */

2024-11-05T20:45:23Z

✏️ Example D1-7.1

← Older revision		Revision as of 20:45, 5 November 2024
Line 296:		Line 296:
	::[[File:ExD1-7-1-eng.png\|thumb\|left\|250px\|link=]]		::[[File:ExD1-7-1-eng.png\|thumb\|left\|250px\|link=]]
	<small>		<small>
	:: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively.		:: Two people, modeled as two particles '''P''' and '''Q''' with the same mass m, are at rest relative to a rotating platform and the ground, respectively.

	:: If '''Q''' studies the <u>dynamics of '''P''' from the ground reference</u>, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.		:: If '''Q''' studies the <u>dynamics of '''P''' from the ground reference</u>, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.

Apons: /* ✏️ Example D1-7.1 */

2024-11-05T20:43:59Z

✏️ Example D1-7.1

← Older revision		Revision as of 20:43, 5 November 2024
Line 298:		Line 298:
	:: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively.		:: Two people, modeled as two particles P and Q with the same mass m, are at rest relative to a rotating platform and the ground, respectively.

	::If '''Q''' studies the <u>dynamics of '''P''' from the ground reference</u>, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.		:: If '''Q''' studies the <u>dynamics of '''P''' from the ground reference</u>, she will not need to include inertia forces because, for this type of problem, the Earth is considered a Galilean reference frame.

	::If '''P''' studies the <u>dynamics of '''Q''' from the rotating platform reference frame</u> (which has the centre '''O''' fixed to the ground and rotates with constant angular velocity <math>\Omega_0</math>), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:		::If '''P''' studies the <u>dynamics of '''Q''' from the rotating platform reference frame</u> (which has the centre '''O''' fixed to the ground and rotates with constant angular velocity <math>\Omega_0</math>), as it is a non-Galilean reference frame (it does not have a rectilinear and uniform translational motion relative to the ground), she will have to include two inertial forces:

	* ● Inertia transportation force <math>\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}</math>: as the Q <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''transportation motion''']]</span> is circular with centre '''O''', radius R (<math>=\mid \OQvec \mid</math>) and associated angular velocity <math>\Omega_0</math>, the transportation force will be radial and outwards (centrifugal) with value <math>\ms\Rs\Omega_0^2</math>.		* Inertia transportation force <math>\mathbf{F}_{\mathrm{ar}\rightarrow \mathrm{Q}}^{-}</math>: as the Q <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''transportation motion''']]</span> is circular with centre '''O''', radius R (<math>=\mid \OQvec \mid</math>) and associated angular velocity <math>\Omega_0</math>, the transportation force will be radial and outwards (centrifugal) with value <math>\ms\Rs\Omega_0^2</math>.

	* ● Coriolis inertia force : it is obtained from the <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''Coriolis acceleration''']]</span>:		* Coriolis inertia force : it is obtained from the <span style="text-decoration: underline;">[[C3. Composition of movements#C3.2 Composition of accelerations\|'''Coriolis acceleration''']]</span>:
	<math>\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})</math><br>		<math>\mathbf{F}_{\text {Cor } \rightarrow \mathbf{Q}}^{-}=-\mathrm{m} \overline{\mathbf{a}}_{\text {Cor }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_{\mathrm{T}}^{\text {Plat }} \times \overline{\mathbf{v}}_{\text {Plat }}(\mathbf{Q})=-\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times\left[\overline{\mathbf{v}}_{\mathrm{T}}(\mathbf{Q})-\overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})\right]=\mathrm{m} 2 \overline{\boldsymbol{\Omega}}_0 \times \overline{\mathbf{v}}_{\mathrm{ar}}(\mathbf{Q})</math><br>