A body resting on a rough horizontal plane require a pull of 18N inclined at 30° to the plane first to move it.It was found that a push of 22N inclined at 30° to the plane just moved the body. Determine the weight and coefficient of friction.


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Question Number 27128 by NECx last updated on 02/Jan/18

$A b o d y r e s t i n g o n a r o u g h$ $h o r i z o n t a l p l a n e r e q u i r e a p u l l o f$ $18 N i n c l i n e d a t 30 ° t o t h e p l a n e$ $f i r s t t o m o v e i t . I t w a s f o u n d$ $t h a t a p u s h o f 22 N i n c l i n e d a t 30 °$ $t o t h e p l a n e j u s t m o v e d t h e b o d y .$ $D e t e r m i n e t h e w e i g h t a n d$ $c o e f f i c i e n t o f f r i c t i o n .$
	Answered by mrW1 last updated on 02/Jan/18

	$P u l l : F_{1} \cos α_{1} = μ (m g - F_{1} \sin α_{1}) . . . (i)$ $P u s h : F_{2} \cos α_{2} = μ (m g + F_{2} \sin α_{2}) . . . (i i)$ $(i) / (i i) :$ $\Rightarrow \frac{F_{1} \cos α_{1}}{F_{2} \cos α_{2}} = \frac{m g - F_{1} \sin α_{1}}{m g + F_{2} \sin α_{2}}$ $\Rightarrow \frac{F_{1} \cos α_{1}}{F_{2} \cos α_{2}} = \frac{m g + F_{2} \sin α_{2} - F_{1} \sin α_{1} - F_{2} \sin α_{2}}{m g + F_{2} \sin α_{2}}$ $\Rightarrow \frac{F_{1} \cos α_{1}}{F_{2} \cos α_{2}} = 1 - \frac{F_{1} \sin α_{1} + F_{2} \sin α_{2}}{m g + F_{2} \sin α_{2}}$ $\Rightarrow 1 - \frac{F_{1} \cos α_{1}}{F_{2} \cos α_{2}} = \frac{F_{1} \sin α_{1} + F_{2} \sin α_{2}}{m g + F_{2} \sin α_{2}}$ $\Rightarrow m g = \frac{F_{1} \sin α_{1} + F_{2} \sin α_{2}}{1 - \frac{F_{1} \cos α_{1}}{F_{2} \cos α_{2}}} - F_{2} \sin α_{2}$ $\Rightarrow m g = F_{2} \sin α_{2} (\frac{1 + \frac{F_{1} \sin α_{1}}{F_{2} \sin α_{2}}}{1 - \frac{F_{1} \cos α_{1}}{F_{2} \cos α_{2}}} - 1)$ $α_{1} = α_{2} = 30 °$ $\Rightarrow m g = 22 \times \frac{1}{2} (\frac{1 + \frac{18}{22}}{1 - \frac{18}{22}} - 1) = 11 \times 9 = 99 N$ $(i i) - (i) :$ $F_{2} \cos α_{2} - F_{1} \cos α_{1} = μ (F_{2} \sin α_{2} + F_{1} \sin α_{1})$ $\Rightarrow μ = \frac{F_{2} \cos α_{2} - F_{1} \cos α_{1}}{F_{2} \sin α_{2} + F_{1} \sin α_{1}}$ $\Rightarrow μ = \frac{22 - 18}{22 + 18} \times \frac{\cos 30 °}{\sin 30 °} = \frac{\sqrt{3}}{10} = 0.173$
	Commented by NECx last updated on 02/Jan/18

	$p l e a s e c a n y o u p o s t a d i a g r a m f o r$ $b e t t e r u n d e r s t a n d i n g . T h i s i s$ $b e c a u s e I n e v e r k n e w t h a t t h e r e$ $w a s a d i f f e r e n c e b e t w e e n t h e r e a c t i o n$ $o f f o r c e s w h e n i n a p u s h a n d p u l l .$
	Commented by mrW1 last updated on 02/Jan/18

	Commented by NECx last updated on 02/Jan/18

	$p l e a s e t h e d i a g r a m y o u d i s p l a y e d$ $i s n o t s h o w i n g p l e a s e c a n y o u p o s t$ $i t a s a q u e s t i o n .$
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