what are the conditions necessary for a body of mass mkg on an incline plane at angle θ to the horizontal to remain at rest?


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Question Number 23022 by NECx last updated on 25/Oct/17

$w h a t a r e t h e c o n d i t i o n s n e c e s s a r y$ $f o r a b o d y o f m a s s m k g o n a n i n c l i n e p l a n e$ $a t a n g l e θ t o t h e h o r i z o n t a l t o$ $r e m a i n a t r e s t ?$
Commented by NECx last updated on 25/Oct/17

$p l e a s e s t a t e a l l n e c e s s a r y$ $c o n d i t i o n s w i t h d i a g r a m s w h e r e$ $n e c e s s a r y . T h a n k s$
Commented by mrW1 last updated on 25/Oct/17

Commented by mrW1 last updated on 25/Oct/17

$1)$ $mg \sin θ ⩽ μ mg \cos θ$ $i . e . \tan θ ⩽ μ$ $2)$ $COM of the body lies behind the out$ $most points of the contact area,$ $i . e . e ⩾ 0$
Commented by NECx last updated on 25/Oct/17

$w o w . . . T h a n k s$
Commented by ajfour last updated on 25/Oct/17

$\vec{N} + \vec{f} + {(\vec{F})}_{e x t} = - m \vec{g}$ $a n d i n i t i a l v e l o c i t y \overset{\to_{}}{u} = 0, a n d$ ${\vec{τ}}_{N} + \vec{τ_{f}} = 0 a b o u t c e n t r e o f m a s s$ $w i t h ω_{0} = 0 .$
	Answered by Physics lover last updated on 25/Oct/17
	$there can be three possibilities : 1. force for friction can balance the component of gravity. { μ ≥ tan θ } 2. there can be an external force on the body { f_(ext ) ≥ mg .Sin θ } 3. the inclined plane/wedge is accelerating and the pseudo force balances the component of gravity. { ma.Cos θ ≥ mg .Sin θ } In All the three cases the body will be at rest w.r.t the surface of the inclined .$
	$t h e r e c a n b e t h r e e p o s s i b i l i t i e s :$ $1 . f o r c e f o r f r i c t i o n c a n b a l a n c e$ $t h e c o m p o n e n t o f g r a v i t y .$ ${μ ⩾ t a n θ}$ $2 . t h e r e c a n b e a n e x t e r n a l f o r c e$ $o n t h e b o d y$ ${f_{e x t} ⩾ m g . S i n θ}$ $3 . t h e i n c l i n e d p l a n e / w e d g e$ $i s a c c e l e r a t i n g a n d t h e p s e u d o$ $f o r c e b a l a n c e s t h e c o m p o n e n t$ $o f g r a v i t y .$ ${m a . C o s θ ⩾ m g . S i n θ}$ $I n A l l t h e t h r e e c a s e s t h e b o d y$ $w i l l b e a t r e s t w . r . t t h e s u r f a c e$ $o f t h e i n c l i n e d .$
	Commented by Physics lover last updated on 25/Oct/17

	Commented by Physics lover last updated on 25/Oct/17

	Commented by Physics lover last updated on 25/Oct/17

	Commented by NECx last updated on 25/Oct/17

	$b u d d y y o u r e x p l a n a t i o n s a r e$ $v i v i d . T h a n k s$
	Commented by Joel577 last updated on 25/Oct/17

	$v e r y n i c e$
	Commented by Physics lover last updated on 25/Oct/17

	$l o l t h a n k s a n d y a r w e l c o m e .$
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