Sliding motion along an inclined plane - Revision history

Vipul: /* Kinematics */

2012-01-16T03:22:46Z

Kinematics

← Older revision		Revision as of 03:22, 16 January 2012
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	\| Time taken on return journey \|\| <math>\! \frac{u}{g \cos \theta\sqrt{\tan^2 \theta - \mu_k^2}}</math>		\| Time taken on return journey \|\| <math>\! \frac{u}{g \cos \theta\sqrt{\tan^2 \theta - \mu_k^2}}</math>
	\|-		\|-
	\| Speed at instant of return \|\| <math>\! u \frac{\tan \theta - \mu_k}{\tan \theta + \mu_k}</math>		\| Speed at instant of return \|\| <math>\! u \sqrt{\frac{\tan \theta - \mu_k}{\tan \theta + \mu_k}}</math>
	\|}		\|}

Vipul at 18:03, 5 December 2011

2011-12-05T18:03:12Z

← Older revision		Revision as of 18:03, 5 December 2011
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			{{perspectives}}
	{{mechanics scenario}}		{{mechanics scenario}}

Vipul: /* External links */

2011-12-05T18:02:35Z

External links

← Older revision		Revision as of 18:02, 5 December 2011
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	===Instructional video links===		===Instructional video links===

	Note that these instructional videos are ''not'' part of Mech and the copyright on these is not with Mech contributors. Please visit the video links in the second column to get more information about the creators and context of the videos.		See [[Video:Sliding motion along an inclined plane]].
	~~{\| class="sortable" border="1"~~
	~~! Video embedded in this page (click SHOW MORE to view) !! Video link (on course page) !!~~ Video ~~link to correct start time (not necessarily on course page) !! Segment !! Contextual information !! Transcript link !! Transcript segment~~
	\|-
	~~\| <toggledisplay>{{#widget~~:~~YouTube\|id=TC23wD34C7k}}</toggledisplay> \|\| [http://www.khanacademy.org/video/~~inclined-plane~~-force-components?playlist=Physics Khan Academy video: inclined plane force components~~] ~~(click through to view this and more videos in the playlist on inclined planes) \|\| \|\| full video \|\| \|\| \|\|~~
	\|-
	\| <toggledisplay>{{#widget:YouTube\|id=uZGbtK2KBoY}}</toggledisplay> \|\| [http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/detail/embed08.htm MIT OCW lecture on friction by Walter Lewin] \|\| \|\| 02:23 - 07:01 \|\| Relates the [[limiting coefficient of static friction]] to the [[angle of friction]~~] to set the stage for some experimental demonstrations. \|\| same as video link \|\| It is fairly easy to measure ... to ...same kind of rubber~~.
	\|}

Vipul: /* Instructional video links */

2011-12-05T17:18:57Z

Instructional video links

← Older revision		Revision as of 17:18, 5 December 2011
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	! Video embedded in this page (click SHOW MORE to view) !! Video link (on course page) !! Video link to correct start time (not necessarily on course page) !! Segment !! Contextual information !! Transcript link !! Transcript segment		! Video embedded in this page (click SHOW MORE to view) !! Video link (on course page) !! Video link to correct start time (not necessarily on course page) !! Segment !! Contextual information !! Transcript link !! Transcript segment
	\|-		\|-
	\| <toggledisplay>{{#widget:YouTube\|id=~~1g6eQqZgc50~~}}</toggledisplay> \|\| [http://www.khanacademy.org/video/~~friction~~-on-an-~~inclined-plane~~ Khan Academy video: ~~friction on an~~ inclined plane] \|\| \|\| full video \|\| \|\| \|\|		\| <toggledisplay>{{#widget:YouTube\|id=TC23wD34C7k}}</toggledisplay> \|\| [http://www.khanacademy.org/video/inclined-plane-force-components?playlist=Physics Khan Academy video: inclined plane force components] (click through to view this and more videos in the playlist on inclined planes) \|\| \|\| full video \|\| \|\| \|\|
	\|-		\|-
	\| <toggledisplay>{{#widget:YouTube\|id=uZGbtK2KBoY}}</toggledisplay> \|\| [http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/detail/embed08.htm MIT OCW lecture on friction by Walter Lewin] \|\| \|\| 02:23 - 07:01 \|\| Relates the [[limiting coefficient of static friction]] to the [[angle of friction]] to set the stage for some experimental demonstrations. \|\| same as video link \|\| It is fairly easy to measure ... to ...same kind of rubber.		\| <toggledisplay>{{#widget:YouTube\|id=uZGbtK2KBoY}}</toggledisplay> \|\| [http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/detail/embed08.htm MIT OCW lecture on friction by Walter Lewin] \|\| \|\| 02:23 - 07:01 \|\| Relates the [[limiting coefficient of static friction]] to the [[angle of friction]] to set the stage for some experimental demonstrations. \|\| same as video link \|\| It is fairly easy to measure ... to ...same kind of rubber.
	\|}		\|}

Vipul at 15:54, 24 August 2011

2011-08-24T15:54:14Z

← Older revision		Revision as of 15:54, 24 August 2011
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	\| Block sliding up the inclined plane \|\| measured positive down the inclined plane (note that acceleration opposes direction of motion, leading to retardation) \|\| <math>\! ma = mg \sin \theta + f_k</math> \|\| <math>\! f_k = \mu_kN</math> \|\| <math>\! ma = mg \sin \theta + \mu_kN</math> \|\| <math>\! a = g (\sin \theta + \mu_k \cos \theta) = g \cos \theta (\tan \theta + \mu_k)</math>		\| Block sliding up the inclined plane \|\| measured positive down the inclined plane (note that acceleration opposes direction of motion, leading to retardation) \|\| <math>\! ma = mg \sin \theta + f_k</math> \|\| <math>\! f_k = \mu_kN</math> \|\| <math>\! ma = mg \sin \theta + \mu_kN</math> \|\| <math>\! a = g (\sin \theta + \mu_k \cos \theta) = g \cos \theta (\tan \theta + \mu_k)</math>
	\|}		\|}
			==What about the inclined plane?==

			A natural question that might occur at this stage is: what is the [[force diagram]] of the inclined plane? And, what is the effect on the inclined plane of the normal force and friction force exerted by the block on it?

			The key thing to note is that by assumption, the inclined plane is ''fixed''. This means that ''whatever mechanism is being used to fix the inclined plane'' is generating the necessary forces to balance the forces exerted by the block, so that by [[Newton's first law of motion]], the inclined plane does not move. For instance, if the inclined plane is fixed to the floor with screws, then the [[contact force]] exerted from the floor ([[normal force]] or friction) exactly balances all the other forces on the inclined plane (including its own [[gravitational force]] and the [[contact force]] from the block).

	==Value of the acceleration function==		==Value of the acceleration function==
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	<math>\! a = g\sqrt{1 + \mu_k^2}</math>		<math>\! a = g\sqrt{1 + \mu_k^2}</math>

	==Behavior for a block initially placed at rest==		==Behavior for a block initially placed at rest==

Vipul: /* Component along (down) the inclined plane */

2011-08-13T17:27:57Z

Component along (down) the inclined plane

← Older revision		Revision as of 17:27, 13 August 2011
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	\| Block stationary \|\| no acceleration, so no sign convention \|\| <math>\! mg \sin \theta = f_s</math> \|\| <math>f_s \le \mu_sN</math> \|\| <math>\! mg \sin \theta \le \mu_sN</math> \|\| <math>\! \tan \theta \le \mu_s</math>		\| Block stationary \|\| no acceleration, so no sign convention \|\| <math>\! mg \sin \theta = f_s</math> \|\| <math>f_s \le \mu_sN</math> \|\| <math>\! mg \sin \theta \le \mu_sN</math> \|\| <math>\! \tan \theta \le \mu_s</math>
	\|-		\|-
	\| Block sliding down the inclined plane \|\| measured positive down the inclined plane, hence a positive number \|\| <math>\! ma = mg \sin \theta - f_k</math> \|\| <math>f_k = \mu_kN</math><math>\! ma = mg \sin \theta - \mu_kN </math> \|\| <math>\! a = g (\sin \theta - \mu_k\cos \theta) = g \cos \theta (\tan \theta - \mu_k)</math>		\| Block sliding down the inclined plane \|\| measured positive down the inclined plane, hence a positive number \|\| <math>\! ma = mg \sin \theta - f_k</math> \|\| <math>\! f_k = \mu_kN</math> \|\| <math>\! ma = mg \sin \theta - \mu_kN </math> \|\| <math>\! a = g (\sin \theta - \mu_k\cos \theta) = g \cos \theta (\tan \theta - \mu_k)</math>
	\|-		\|-
	\| Block sliding up the inclined plane \|\| measured positive down the inclined plane (note that acceleration opposes direction of motion, leading to retardation) \|\| <math>\! ma = mg \sin \theta + f_k</math> \|\| <math>f_k = \mu_kN</math> \|\| <math>\! ma = mg \sin \theta + \mu_kN</math> \|\| <math>\! a = g (\sin \theta + \mu_k \cos \theta) = g \cos \theta (\tan \theta + \mu_k)</math>		\| Block sliding up the inclined plane \|\| measured positive down the inclined plane (note that acceleration opposes direction of motion, leading to retardation) \|\| <math>\! ma = mg \sin \theta + f_k</math> \|\| <math>\! f_k = \mu_kN</math> \|\| <math>\! ma = mg \sin \theta + \mu_kN</math> \|\| <math>\! a = g (\sin \theta + \mu_k \cos \theta) = g \cos \theta (\tan \theta + \mu_k)</math>
	\|}		\|}

Vipul: /* Component along (down) the inclined plane */

2011-08-13T17:27:02Z

Component along (down) the inclined plane

← Older revision		Revision as of 17:27, 13 August 2011
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	===Component along (down) the inclined plane===		===Component along (down) the inclined plane===

	For the axis ''down'' the inclined plane, the gravitational force component is <math>mg\sin \theta</math>. The magnitude and direction of the friction force are not clear. We have three cases, the case of ''no sliding'', where an upward static friction <math>f_s</math> acts, the case of ''sliding down'', where an upward kinetic friction <math>f_k = \mu_kN</math> acts, and the case of ''sliding up'', where a downward kinetic friction <math>f_k = \mu_kN</math> acts.		For the axis ''down'' the inclined plane, the gravitational force component is <math>mg\sin \theta</math>. The magnitude and direction of the friction force are not clear. We have three cases, the case of ''no sliding'', where an upward static friction <math>f_s</math> acts (subject to the constraint <math>f_s \le \mu_sN</math>), the case of ''sliding down'', where an upward kinetic friction <math>f_k = \mu_kN</math> acts, and the case of ''sliding up'', where a downward kinetic friction <math>f_k = \mu_kN</math> acts.
	:		:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Case !! Convention on the sign of acceleration <math>a</math> !! Equation using <math>f_s, f_k</math> !! Simplified, substituting <math>f_s,f_k</math> in terms of <math>N,\mu_s,\mu_k</math> !! Simplified, after combining with <math>\! N = mg \cos \theta</math>		! Case !! Convention on the sign of acceleration <math>a</math> !! Equation using <math>f_s, f_k</math> !! Constraint on <math>f_s, f_k</math> !! Simplified, substituting <math>f_s,f_k</math> in terms of <math>N,\mu_s,\mu_k</math> !! Simplified, after combining with <math>\! N = mg \cos \theta</math>
	\|-		\|-
	\| Block stationary \|\| no acceleration, so no sign convention \|\| <math>\! mg \sin \theta = f_s</math> \|\| <math>\! mg \sin \theta \le \mu_sN</math> \|\| <math>\! \tan \theta \le \mu_s</math>		\| Block stationary \|\| no acceleration, so no sign convention \|\| <math>\! mg \sin \theta = f_s</math> \|\| <math>f_s \le \mu_sN</math> \|\| <math>\! mg \sin \theta \le \mu_sN</math> \|\| <math>\! \tan \theta \le \mu_s</math>
	\|-		\|-
	\| Block sliding down the inclined plane \|\| measured positive down the inclined plane, hence a positive number \|\| <math>\! ma = mg \sin \theta - f_k</math> \|\| <math>\! ma = mg \sin \theta - \mu_kN </math> \|\| <math>\! a = g (\sin \theta - \mu_k\cos \theta) = g \cos \theta (\tan \theta - \mu_k)</math>		\| Block sliding down the inclined plane \|\| measured positive down the inclined plane, hence a positive number \|\| <math>\! ma = mg \sin \theta - f_k</math> \|\| <math>f_k = \mu_kN</math><math>\! ma = mg \sin \theta - \mu_kN </math> \|\| <math>\! a = g (\sin \theta - \mu_k\cos \theta) = g \cos \theta (\tan \theta - \mu_k)</math>
	\|-		\|-
	\| Block sliding up the inclined plane \|\| measured positive down the inclined plane (note that acceleration opposes direction of motion, leading to retardation) \|\| <math>\! ma = mg \sin \theta + f_k</math> \|\| <math>\! ma = mg \sin \theta + \mu_kN</math> \|\| <math>\! a = g (\sin \theta + \mu_k \cos \theta) = g \cos \theta (\tan \theta + \mu_k)</math>		\| Block sliding up the inclined plane \|\| measured positive down the inclined plane (note that acceleration opposes direction of motion, leading to retardation) \|\| <math>\! ma = mg \sin \theta + f_k</math> \|\| <math>f_k = \mu_kN</math> \|\| <math>\! ma = mg \sin \theta + \mu_kN</math> \|\| <math>\! a = g (\sin \theta + \mu_k \cos \theta) = g \cos \theta (\tan \theta + \mu_k)</math>
	\|}		\|}

Vipul at 13:40, 12 August 2011

2011-08-12T13:40:38Z

← Older revision		Revision as of 13:40, 12 August 2011
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	[[File:Blockonincline.png\|thumb\|500px\|right\|The brown triangle is the fixed inclined plane and the black block is placed on it with a dry surface of contact.]]		[[File:Blockonincline.png\|thumb\|500px\|right\|The brown triangle is the fixed inclined plane and the black block is placed on it with a dry surface of contact.]]

	The scenario here is a dry block (with a stable surface of contact) on a dry ''fixed'' [[involves ~~simple machine~~::inclined plane]], with <math>\theta</math> being the angle of inclination with the horizontal axis.		The scenario here is a dry block (with a stable surface of contact) on a dry ''fixed'' [[involves::inclined plane]], with <math>\theta</math> being the angle of inclination with the horizontal axis.

	The extremes are <math>\theta = 0</math> (whence, the plane is horizontal) and <math>\theta = \pi/2</math> (whence, the plane is vertical).		The extremes are <math>\theta = 0</math> (whence, the plane is horizontal) and <math>\theta = \pi/2</math> (whence, the plane is vertical).

Vipul at 13:39, 12 August 2011

2011-08-12T13:39:06Z

← Older revision		Revision as of 13:39, 12 August 2011
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	[[File:Blockonincline.png\|thumb\|500px\|right\|The brown triangle is the fixed inclined plane and the black block is placed on it with a dry surface of contact.]]		[[File:Blockonincline.png\|thumb\|500px\|right\|The brown triangle is the fixed inclined plane and the black block is placed on it with a dry surface of contact.]]

	The scenario here is a dry block (with a stable surface of contact) on a dry ''fixed'' [involves simple machine::inclined plane]], with <math>\theta</math> being the angle of inclination with the horizontal axis.		The scenario here is a dry block (with a stable surface of contact) on a dry ''fixed'' [[involves simple machine::inclined plane]], with <math>\theta</math> being the angle of inclination with the horizontal axis.

	The extremes are <math>\theta = 0</math> (whence, the plane is horizontal) and <math>\theta = \pi/2</math> (whence, the plane is vertical).		The extremes are <math>\theta = 0</math> (whence, the plane is horizontal) and <math>\theta = \pi/2</math> (whence, the plane is vertical).

Vipul at 13:38, 12 August 2011

2011-08-12T13:38:26Z

← Older revision		Revision as of 13:38, 12 August 2011
Line 3:		Line 3:
	[[File:Blockonincline.png\|thumb\|500px\|right\|The brown triangle is the fixed inclined plane and the black block is placed on it with a dry surface of contact.]]		[[File:Blockonincline.png\|thumb\|500px\|right\|The brown triangle is the fixed inclined plane and the black block is placed on it with a dry surface of contact.]]

	The scenario here is a dry block (with a stable surface of contact) on a dry ''fixed'' inclined plane, with <math>\theta</math> being the angle of inclination with the horizontal axis.		The scenario here is a dry block (with a stable surface of contact) on a dry ''fixed'' [involves simple machine::inclined plane]], with <math>\theta</math> being the angle of inclination with the horizontal axis.

	The extremes are <math>\theta = 0</math> (whence, the plane is horizontal) and <math>\theta = \pi/2</math> (whence, the plane is vertical).		The extremes are <math>\theta = 0</math> (whence, the plane is horizontal) and <math>\theta = \pi/2</math> (whence, the plane is vertical).