Fixed pulley with vertical loads - Revision history

Vipul: /* Case of a massless string and a pulley with moment of inertia */

2011-08-12T13:44:47Z

Case of a massless string and a pulley with moment of inertia

← Older revision		Revision as of 13:44, 12 August 2011
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	! Case !! Direction of acceleration !! Magnitude of acceleration in terms of <math>m_1,m_2,I,R</math> !! Magnitude of acceleration assuming pulley is a solid disk of constant density and mass <math>M</math>		! Case !! Direction of acceleration !! Magnitude of acceleration in terms of <math>m_1,m_2,I,R</math> !! Magnitude of acceleration assuming pulley is a solid disk of constant density and mass <math>M</math>
	\|-		\|-
	\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_1 - m_2)g}{(m_1 + m_2 + (M/2)}</math>		\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (M/2)}</math>
	\|-		\|-
	\| <math>m_2 > m_1</math> \|\| upward on <math>m_1</math>, downward on <math>m_2</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (M/2)}</math>		\| <math>m_2 > m_1</math> \|\| upward on <math>m_1</math>, downward on <math>m_2</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (M/2)}</math>

Vipul: /* Case of a massless string and a pulley with moment of inertia */

2010-11-23T00:46:02Z

Case of a massless string and a pulley with moment of inertia

← Older revision		Revision as of 00:46, 23 November 2010
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	\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_1 - m_2)g}{(m_1 + m_2 + (M/2)}</math>		\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_1 - m_2)g}{(m_1 + m_2 + (M/2)}</math>
	\|-		\|-
	\| <math>m_2 > m_1</math> \|\| upward on <math>m_1</math>, downward on <math>m_2</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_2 - m_1)g}{(m_1 + m_2 + (M/2)}</math>		\| <math>m_2 > m_1</math> \|\| upward on <math>m_1</math>, downward on <math>m_2</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (M/2)}</math>
	\|-		\|-
	\| <math>m_1 = m_2</math> \|\| no direction \|\| zero (system in equilibrium) \|\| zero (system in equilibrium)		\| <math>m_1 = m_2</math> \|\| no direction \|\| zero (system in equilibrium) \|\| zero (system in equilibrium)
	\|}		\|}

Vipul: /* Case of a massless string and a pulley with moment of inertia */

2010-11-23T00:45:30Z

Case of a massless string and a pulley with moment of inertia

← Older revision		Revision as of 00:45, 23 November 2010
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Case !! Direction of acceleration !! Magnitude of acceleration in terms of <math>m_1,m_2,I,R</math> !! Magnitude of acceleration assuming pulley is a solid disk of constant density and mass <math>M<math>		! Case !! Direction of acceleration !! Magnitude of acceleration in terms of <math>m_1,m_2,I,R</math> !! Magnitude of acceleration assuming pulley is a solid disk of constant density and mass <math>M</math>
	\|-		\|-
	\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_1 - m_2)g}{(m_1 + m_2 + (M/2)}</math>		\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_1 - m_2)g}{(m_1 + m_2 + (M/2)}</math>

Vipul at 00:45, 23 November 2010

2010-11-23T00:45:03Z

← Older revision		Revision as of 00:45, 23 November 2010
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	{{quotation\|'''KEY FORCE CONCEPT''': There is tension in the string in ''all'' cases, even when the system is in equilibrium and hence neither mass is accelerating. Tension is what keeps the string taut and in place. Tension goes to zero only when one of the masses goes to zero, which means that the string is no longer taut at that end.}}		{{quotation\|'''KEY FORCE CONCEPT''': There is tension in the string in ''all'' cases, even when the system is in equilibrium and hence neither mass is accelerating. Tension is what keeps the string taut and in place. Tension goes to zero only when one of the masses goes to zero, which means that the string is no longer taut at that end.}}

			==Case of a massless string and a pulley with moment of inertia==

			Here's the version in the case that the pulley and the string do not slip against each other. Thus, if <math>a</math> is the acceleration of the string, the angular acceleration of the pulley is <math>\alpha = a/R</math>. Note that this requires static friction operating between the string and the pulley. We denote by <math>I</math> the moment of inertia of the pulley and by <math>R</math> the radius of the pulley.

			{\| class="sortable" border="1"
			! Case !! Direction of acceleration !! Magnitude of acceleration in terms of <math>m_1,m_2,I,R</math> !! Magnitude of acceleration assuming pulley is a solid disk of constant density and mass <math>M<math>
			\|-
			\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>\frac{(m_1 - m_2)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_1 - m_2)g}{(m_1 + m_2 + (M/2)}</math>
			\|-
			\| <math>m_2 > m_1</math> \|\| upward on <math>m_1</math>, downward on <math>m_2</math> \|\| <math>\frac{(m_2 - m_1)g}{m_1 + m_2 + (I/R^2)}</math> \|\| <math>\frac{(m_2 - m_1)g}{(m_1 + m_2 + (M/2)}</math>
			\|-
			\| <math>m_1 = m_2</math> \|\| no direction \|\| zero (system in equilibrium) \|\| zero (system in equilibrium)
			\|}

Vipul at 22:58, 22 November 2010

2010-11-22T22:58:34Z

← Older revision		Revision as of 22:58, 22 November 2010
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	[[File:Fixedpulleywithverticalloads.png\|thumb\|500px\|right]]		[[File:Fixedpulleywithverticalloads.png\|thumb\|500px\|right]]

			==Case of a massless pulley and string==

			In case the pulley and string are massless, the upward tension force exerted by the string on <math>m_1</math> equals the upward tension force exerted by the string on <math>m_2</math>. If we call this value of tension <math>T</math>, then the force diagrams of the two masses are as shown below.

			{{fillin}}

			We have the following cases:

			{\| class="sortable" border="1"
			! Case !! Direction of acceleration !! Magnitude of acceleration !! Magnitude of tension force <math>T</math>
			\|-
			\| <math>m_1 > m_2</math> \|\| upward on <math>m_2</math>, downward on <math>m_1</math> \|\| <math>(m_1 - m_2)g/(m_1 + m_2)</math> \|\| <math>\frac{2m_1m_2g}{m_1 + m_2}</math> -- this is <math>g</math> times the harmonic mean of <math>m_1</math> and <math>m_2</math>
			\|-
			\| <math>m_1 < m_2</math> \|\| downward on <math>m_2</math>, upward on <math>m_1</math> \|\| <math>(m_2 - m_1)g/(m_1 + m_2)</math> \|\| <math>\frac{2m_1m_2g}{m_1 + m_2}</math> -- this is <math>g</math> times the harmonic mean of <math>m_1</math> and <math>m_2</math>
			\|-
			\| <math>m_1 = m_2</math> \|\| no direction \|\| zero \|\| <math>\frac{2m_1m_2g}{m_1 + m_2}</math> -- this is <math>g</math> times the harmonic mean of <math>m_1</math> and <math>m_2</math>
			\|}

			Note that the magnitude of the tension force has the same expression in all cases.

			{{quotation\|'''KEY FORCE CONCEPT''': There is tension in the string in ''all'' cases, even when the system is in equilibrium and hence neither mass is accelerating. Tension is what keeps the string taut and in place. Tension goes to zero only when one of the masses goes to zero, which means that the string is no longer taut at that end.}}

Vipul: Created page with "{{mechanics scenario}} This article describes a special case of a simple pulley system: a fixed pulley with vertical loads connected by a string passing over the pulley, as depi..."

2010-11-22T22:48:50Z

Created page with "{{mechanics scenario}} This article describes a special case of a simple pulley system: a fixed pulley with vertical loads connected by a string passing over the pulley, as depi..."

New page

{{mechanics scenario}}

This article describes a special case of a simple pulley system: a fixed pulley with vertical loads connected by a string passing over the pulley, as depicted in the figure.

[[File:Fixedpulleywithverticalloads.png|thumb|500px|right]]