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Question Number 81562 by ajfour last updated on 14/Feb/20

Commented by ajfour last updated on 14/Feb/20

$I f p u l l e d s l i g h t l y m o r e,$ $h e m i s p h e r e s l i p s, f i n d t e n s i o n$ $i n s h o w n e q u i l i b r i u m s t a t e .$
Commented by mr W last updated on 14/Feb/20

$μ = 0.6 l e a d s t o n o s o l u t i o n . i h a v e$ $s e t μ = 0.2 .$
	Answered by mr W last updated on 14/Feb/20

	Commented by mr W last updated on 15/Feb/20
	$l sin ϕ+R sin θ+R=h 2 sin ϕ+sin θ+1=3 ⇒2 sin ϕ+sin θ=2 ...(i) N+T sin ϕ=mg ...(ii) T cos ϕ=f=μN ...(iii) T cos ϕ (R sin θ+R)+T sin ϕ R cos θ=mg e sin θ T [sin (ϕ+θ)+cos ϕ]=mg e sin θ ...(iv) ⇒sin θ=2(1−sin ϕ) T cos ϕ=μ(mg−T sin ϕ) T(cos ϕ+μ sin ϕ)=μmg ⇒T=((μmg)/(cos ϕ+μ sin ϕ)) ((μmg)/(cos ϕ+μ sin ϕ))[sin (ϕ+θ)+cos ϕ]=mg e sin θ ((μ{sin (ϕ+θ)+cos ϕ})/(cos ϕ+μ sin ϕ))=2e(1−sin ϕ) e=((3R)/8)=(3/8) ⇒((sin (ϕ+θ)+cos ϕ)/((cos ϕ+μ sin ϕ)(1−sin ϕ)))=(3/(4μ)) with θ=sin^(−1) [2(1−sin ϕ)] with μ=0.6 there is no solution possible. let′s say μ=0.2, we get ϕ≈30.3375° ⇒T=155.59 N ===================== μ=((3 cos ϕ(1−sin ϕ))/(4sin (ϕ+θ)+4cos ϕ+3 sin^2 ϕ−3sin ϕ)) ϕ→30° θ→90° μ<(3/(16−(√3)))≈0.21$
	$l \sin φ + R \sin θ + R = h$ $2 \sin φ + \sin θ + 1 = 3$ $\Rightarrow 2 \sin φ + \sin θ = 2 . . . (i)$ $N + T \sin φ = m g . . . (i i)$ $T \cos φ = f = μ N . . . (i i i)$ $T \cos φ (R \sin θ + R) + T \sin φ R \cos θ = m g e \sin θ$ $T [\sin (φ + θ) + \cos φ] = m g e \sin θ . . . (i v)$ $\Rightarrow \sin θ = 2 (1 - \sin φ)$ $T \cos φ = μ (m g - T \sin φ)$ $T (\cos φ + μ \sin φ) = μ m g$ $\Rightarrow T = \frac{μ m g}{\cos φ + μ \sin φ}$ $\frac{μ m g}{\cos φ + μ \sin φ} [\sin (φ + θ) + \cos φ] = m g e \sin θ$ $\frac{μ {\sin (φ + θ) + \cos φ}}{\cos φ + μ \sin φ} = 2 e (1 - \sin φ)$ $e = \frac{3 R}{8} = \frac{3}{8}$ $\Rightarrow \frac{\sin (φ + θ) + \cos φ}{(\cos φ + μ \sin φ) (1 - \sin φ)} = \frac{3}{4 μ}$ $w i t h θ = \sin^{- 1} [2 (1 - \sin φ)]$ $w i t h μ = 0.6 t h e r e i s n o s o l u t i o n p o s s i b l e .$ ${l e t}^{'} s s a y μ = 0.2, w e g e t$ $φ \approx 30.3375 °$ $\Rightarrow T = 155.59 N$ $=====================$ $μ = \frac{3 \cos φ (1 - \sin φ)}{4 \sin (φ + θ) + 4 \cos φ + 3 \sin^{2} φ - 3 \sin φ}$ $φ \to 30 °$ $θ \to 90 °$ $μ < \frac{3}{16 - \sqrt{3}} \approx 0.21$
	Commented by ajfour last updated on 15/Feb/20

	$Y o u h a v e s o l v e d i t i n a v e r y$ $l u c i d w a y s i r! I s e e i t n o w,$ $t h a n k s .$ $(u s e l e s s t r y i n g . T h e w a y y o u$ $s o l v e d i t S i r, i s s a t i s f a c t o r y$ $e n o u g h!)$
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