Pulley system on a double inclined plane - Revision history

Vipul: /* Basic components of force diagram */

2012-04-19T19:56:01Z

Basic components of force diagram

← Older revision		Revision as of 19:56, 19 April 2012
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	! Force (letter) !! Nature of force !! Condition for existence !! Magnitude !! Direction		! Force (letter) !! Nature of force !! Condition for existence !! Magnitude !! Direction
	\|-		\|-
	\| <math>m_1g</math> \|\| [[gravitational force]] \|\| unconditional\|\| <math>m_1g</math> where <math>m_1</math> is the [[mass]] and <math>g</math> is the [[acceleration due to gravity]] \|\| vertically downward, hence an angle <math>\~~theta~~</math> with the normal to the incline and an angle <math>(\pi/2) - \alpha_1</math> with the incline		\| <math>m_1g</math> \|\| [[gravitational force]] \|\| unconditional\|\| <math>m_1g</math> where <math>m_1</math> is the [[mass]] and <math>g</math> is the [[acceleration due to gravity]] \|\| vertically downward, hence an angle <math>\alpha_1</math> with the normal to the incline and an angle <math>(\pi/2) - \alpha_1</math> with the incline
	\|-		\|-
	\| <math>T</math> \|\| [[tension]] \|\| unconditional \|\| needs to be computed based on solving equations. \|\| pulls the mass up the inclined plane.		\| <math>T</math> \|\| [[tension]] \|\| unconditional \|\| needs to be computed based on solving equations. \|\| pulls the mass up the inclined plane.
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	! Force (letter) !! Nature of force !! Condition for existence !! Magnitude !! Direction		! Force (letter) !! Nature of force !! Condition for existence !! Magnitude !! Direction
	\|-		\|-
	\| <math>m_2g</math> \|\| [[gravitational force]] \|\| unconditional\|\| <math>m_2g</math> where <math>m_2</math> is the [[mass]] and <math>g</math> is the [[acceleration due to gravity]] \|\| vertically downward, hence an angle <math>\~~theta~~</math> with the normal to the incline and an angle <math>(\pi/2) - \alpha_2</math> with the incline		\| <math>m_2g</math> \|\| [[gravitational force]] \|\| unconditional\|\| <math>m_2g</math> where <math>m_2</math> is the [[mass]] and <math>g</math> is the [[acceleration due to gravity]] \|\| vertically downward, hence an angle <math>\alpha_2</math> with the normal to the incline and an angle <math>(\pi/2) - \alpha_2</math> with the incline
	\|-		\|-
	\| <math>T</math> \|\| [[tension]] \|\| unconditional \|\| needs to be computed based on solving equations. \|\| pulls the mass up the inclined plane.		\| <math>T</math> \|\| [[tension]] \|\| unconditional \|\| needs to be computed based on solving equations. \|\| pulls the mass up the inclined plane.

Vipul: /* Basic components of force diagram */

2011-09-07T12:16:59Z

Basic components of force diagram

← Older revision		Revision as of 12:16, 7 September 2011
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	\| <math>f_{s1}</math> \|\| [[static friction]] \|\| no sliding \|\| adjusts so that there is no net acceleration along the incline. At most equal to <math>\mu_{s1}N_1</math>. \|\| The direction could be either up or down the incline, depending on whether the remaining forces create a net tendency to pull <math>m_1</math> down or pull it up.		\| <math>f_{s1}</math> \|\| [[static friction]] \|\| no sliding \|\| adjusts so that there is no net acceleration along the incline. At most equal to <math>\mu_{s1}N_1</math>. \|\| The direction could be either up or down the incline, depending on whether the remaining forces create a net tendency to pull <math>m_1</math> down or pull it up.
	\|-		\|-
	\| <math>f_{k1}</math> \|\| [[kinetic friction]] \|\| sliding \|\| <math>\mu_{k1}N_1</math> where <math>N_1</math> is the normal force \|\| opposite direction of motion -- hence down the incline if sliding up, up the incline if sliding down~~. Assuming the block was ''initially'' at rest, it can only slide down, hence ''up the incline'' is the only possibility~~.		\| <math>f_{k1}</math> \|\| [[kinetic friction]] \|\| sliding \|\| <math>\mu_{k1}N_1</math> where <math>N_1</math> is the normal force \|\| opposite direction of motion -- hence down the incline if sliding up, up the incline if sliding down.
	\|}		\|}

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	\| <math>f_{s2}</math> \|\| [[static friction]] \|\| no sliding \|\| adjusts so that there is no net acceleration along the incline. At most equal to <math>\mu_{s2}N_2</math>. \|\| The direction could be either up or down the incline, depending on whether the remaining forces create a net tendency to pull <math>m_2</math> down or pull it up.		\| <math>f_{s2}</math> \|\| [[static friction]] \|\| no sliding \|\| adjusts so that there is no net acceleration along the incline. At most equal to <math>\mu_{s2}N_2</math>. \|\| The direction could be either up or down the incline, depending on whether the remaining forces create a net tendency to pull <math>m_2</math> down or pull it up.
	\|-		\|-
	\| <math>f_{k2}</math> \|\| [[kinetic friction]] \|\| sliding \|\| <math>\mu_{k2}N_2</math> where <math>N_2</math> is the normal force \|\| opposite direction of motion -- hence down the incline if sliding up, up the incline if sliding down~~. Assuming the block was ''initially'' at rest, it can only slide down, hence ''up the incline'' is the only possibility~~.		\| <math>f_{k2}</math> \|\| [[kinetic friction]] \|\| sliding \|\| <math>\mu_{k2}N_2</math> where <math>N_2</math> is the normal force \|\| opposite direction of motion -- hence down the incline if sliding up, up the incline if sliding down.
	\|}		\|}

Vipul: /* Basic components of force diagram */

2011-08-14T01:30:17Z

Basic components of force diagram

← Older revision		Revision as of 01:30, 14 August 2011
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	\|}		\|}

	A similar description is valid for math>m_2</math>:		A similar description is valid for <math>m_2</math>:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Summary of cases starting from rest */

2011-08-14T01:29:53Z

Summary of cases starting from rest

← Older revision		Revision as of 01:29, 14 August 2011
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	! Case !! What happens qualitatively !! Magnitude of accelerations		! Case !! What happens qualitatively !! Magnitude of accelerations
	\|-		\|-
	\| <math>\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_1</math> slides downward and <math>m_2</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g(m_1 \sin \alpha_1 - m_2 \sin \alpha_2 - \mu_{k1}m_1\~~sin~~ \alpha_1 - \mu_{k2}m_2 \~~sin~~\alpha_2)/(m_1 + m_2)</math>.		\| <math>\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_1</math> slides downward and <math>m_2</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g(m_1 \sin \alpha_1 - m_2 \sin \alpha_2 - \mu_{k1}m_1\cos\alpha_1 - \mu_{k2}m_2 \cos\alpha_2)/(m_1 + m_2)</math>.
	\|-		\|-
	\| <math>\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_2</math> slides downward and <math>m_1</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g(m_2 \sin \alpha_2 - m_1 \sin \alpha_1 - \mu_{k1}m_1\~~sin~~ \alpha_1 - \mu_{k2}m_2 \~~sin~~\alpha_2)/(m_1 + m_2)</math>.		\| <math>\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_2</math> slides downward and <math>m_1</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g(m_2 \sin \alpha_2 - m_1 \sin \alpha_1 - \mu_{k1}m_1\cos \alpha_1 - \mu_{k2}m_2 \cos\alpha_2)/(m_1 + m_2)</math>.
	\|-		\|-
	\| <math>\! \|m_1 \sin \alpha_1 - m_2 \sin \alpha_2\| \le \|\mu_{s1}m_1 \cos \alpha_1 + \mu_{s2}m_2 \cos \alpha_2\|</math> \|\| The system remains at rest \|\| <math>0</math>		\| <math>\! \|m_1 \sin \alpha_1 - m_2 \sin \alpha_2\| \le \|\mu_{s1}m_1 \cos \alpha_1 + \mu_{s2}m_2 \cos \alpha_2\|</math> \|\| The system remains at rest \|\| <math>0</math>

Vipul at 01:29, 14 August 2011

2011-08-14T01:29:03Z

Show changes

Vipul at 00:07, 13 August 2011

2011-08-13T00:07:08Z

← Older revision		Revision as of 00:07, 13 August 2011
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	This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal, thus making it a [[involves::double inclined plane]]. A [[involves::pulley]] is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.		This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal, thus making it a [[involves::double inclined plane]]. A [[involves::pulley]] is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.

			We assume the pulley to be massless so that its moment of inertia can be ignored for the information below.

	==Summary of cases starting from rest==		==Summary of cases starting from rest==

Vipul at 13:42, 12 August 2011

2011-08-12T13:42:28Z

← Older revision		Revision as of 13:42, 12 August 2011
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	[[File:Pulleysystemondoubleinclinedplane.png\|thumb\|400px\|right]]		[[File:Pulleysystemondoubleinclinedplane.png\|thumb\|400px\|right]]

	This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal. A pulley is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.		This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal, thus making it a [[involves::double inclined plane]]. A [[involves::pulley]] is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.

	==Summary of cases starting from rest==		==Summary of cases starting from rest==

Vipul: /* Summary of cases starting from rest */

2010-09-18T18:13:33Z

Summary of cases starting from rest

← Older revision		Revision as of 18:13, 18 September 2010
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	! Case !! What happens qualitatively !! Magnitude of accelerations		! Case !! What happens qualitatively !! Magnitude of accelerations
	\|-		\|-
	\| <math>\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_1</math> slides downward and <math>m_2</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g[m_1 \sin \alpha_1 - m_2 \sin \alpha_2 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2]</math>.		\| <math>\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_1</math> slides downward and <math>m_2</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g(m_1 \sin \alpha_1 - m_2 \sin \alpha_2 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2)/(m_1 + m_2)</math>.
	\|-		\|-
	\| <math>\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_2</math> slides downward and <math>m_1</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g[m_2 \sin \alpha_2 - m_1 \sin \alpha_1 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2]</math>.		\| <math>\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> \|\| <math>m_2</math> slides downward and <math>m_1</math> slides upward, with the same magnitude of acceleration \|\| <math>\! a = g(m_2 \sin \alpha_2 - m_1 \sin \alpha_1 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2)/(m_1 + m_2)</math>.
	\|-		\|-
	\| <math>\! \|m_1 \sin \alpha_1 - m_2 \sin \alpha_2\| \le \|\mu_{s1}m_1 \cos \alpha_1 + \mu_{s2}m_2 \cos \alpha_2\|</math> \|\| The system remains at rest \|\| <math>0</math>		\| <math>\! \|m_1 \sin \alpha_1 - m_2 \sin \alpha_2\| \le \|\mu_{s1}m_1 \cos \alpha_1 + \mu_{s2}m_2 \cos \alpha_2\|</math> \|\| The system remains at rest \|\| <math>0</math>
	\|}		\|}

Vipul at 01:12, 17 September 2010

2010-09-17T01:12:25Z

← Older revision		Revision as of 01:12, 17 September 2010
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	{{mechanics scenario}}		{{mechanics scenario}}

			[[File:Pulleysystemondoubleinclinedplane.png\|thumb\|400px\|right]]

	This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal. A pulley is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.		This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal. A pulley is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.

Vipul: Created page with '{{mechanics scenario}} This article is about the following scenario. A fixed triangular wedge has two inclines $I_1$ and $I_2$ making angles $\alpha_1…'$

2010-06-27T16:23:56Z

Created page with '{{mechanics scenario}} This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1…'

New page

{{mechanics scenario}}

This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal. A pulley is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.

==Summary of cases starting from rest==

{| class="sortable" border="1"
! Case !! What happens qualitatively !! Magnitude of accelerations
|-
| <math>\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> || <math>m_1</math> slides downward and <math>m_2</math> slides upward, with the same magnitude of acceleration || <math>\! a = g[m_1 \sin \alpha_1 - m_2 \sin \alpha_2 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2]</math>.
|-
| <math>\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> || <math>m_2</math> slides downward and <math>m_1</math> slides upward, with the same magnitude of acceleration || <math>\! a = g[m_2 \sin \alpha_2 - m_1 \sin \alpha_1 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2]</math>.
|-
| <math>\! |m_1 \sin \alpha_1 - m_2 \sin \alpha_2| \le |\mu_{s1}m_1 \cos \alpha_1 + \mu_{s2}m_2 \cos \alpha_2|</math> || The system remains at rest || <math>0</math>
|}

Pulley system on a double inclined plane - Revision history

Vipul: /* Basic components of force diagram */

Vipul: /* Basic components of force diagram */

Vipul: /* Basic components of force diagram */

Vipul: /* Summary of cases starting from rest */

Vipul at 01:29, 14 August 2011

Vipul at 00:07, 13 August 2011

Vipul at 13:42, 12 August 2011

Vipul: /* Summary of cases starting from rest */

Vipul at 01:12, 17 September 2010

Vipul: Created page with '{{mechanics scenario}} This article is about the following scenario. A fixed triangular wedge has two inclines I_1 and I_2 making angles \alpha_1…'

Vipul: Created page with '{{mechanics scenario}} This article is about the following scenario. A fixed triangular wedge has two inclines $I_1$ and $I_2$ making angles $\alpha_1…'$