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Physics

Mechanics

Easy

Question

A spool of mass M and internal and external radii R and 2R hanging from a rope touches a curved surface, as shown. A block of mass m placed on a rough surface inclined at an angle a with horizontal is attached with other end of the rope. The pulley is massless and system is in equilibrium. Find the coefficient of friction

$\frac{3 m g + 2 M g}{3 m g - 2 M g}$
$\frac{3 m g \sin α + 2 M g \cos α}{3 m g \cos α - 2 M g \sin α}$
$\frac{3 m g \cos α + 2 M g \sin α}{3 m g \sin α - 2 M g \cos α}$
$\frac{3 m g + 2 M g \tan α}{3 m g - 2 M g \tan α}$

The correct answer is: $fraction numerator 3 m g sin invisible function application alpha plus 2 M g cos invisible function application alpha over denominator 3 m g cos invisible function application alpha minus 2 M g sin invisible function application alpha end fraction$

$T o r q u e space a b o u t space p o i n t space o f space c o n t a c t space o f space t h e space s p o o l space w i l l space b e space z e r o. T.3 R equals M g.2 R space of 1em text or end text T equals 2 over 3 M g E q u a t i n g space t h e space f o r c e s space a c t i n g space o n space t h e space b l o c k space a l o n g space a n d space p e r p e n d i c u l a r space t o space t h e space i n c l i n e table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell N plus T sin invisible function application alpha equals m g cos invisible function application alpha end cell row cell text or end text N equals m g cos invisible function application alpha minus 2 over 3 M g sin invisible function application alpha end cell row cell text Also end text m g sin invisible function application alpha plus T cos invisible function application alpha equals f equals mu N end cell row cell m g sin invisible function application alpha plus 2 over 3 M g cos invisible function application alpha equals mu open square brackets m g cos invisible function application alpha minus 2 over 3 M g sin invisible function application alpha close square brackets end cell end table mu equals fraction numerator 3 m g sin invisible function application alpha plus 2 M g cos invisible function application alpha over denominator 3 m g cos invisible function application alpha minus 2 M g sin invisible function application alpha end fraction$

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$text A linear impulse end text integral F d t text acts at a point end text straight C text of the smooth end text rod invisible function application AB text . The value of end text straight x text is so that the end end text A space r e m a i n s space s t a t i o n a r y space j u s t space a f t e r space t h e space i m p a c t space i s space colon$

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