A massless spool of inner radius r , outer radius R is placed against vertical wall and tilted split floor as shown. A light inextensible thread is tightly wound around the spool through which a mass

Question

A massless spool of inner radius r , outer radius R is placed against vertical wall and tilted split floor as shown. A light inextensible thread is tightly wound around the spool through which a mass M is hanging. There exists no friction at point A , while the coefficient of friction between spool and point B is μ . The angle between two surfaces is θ .

TestHub · Accepted Answer

mgr = fR N 1 ​ sin q + f = mg N 1 ​ cos q = N 2 ​ from (i), f = R mgr ​ From (ii) and (iii), N 2 ​ = tan θ mg − f ​ = tan θ mg ​ [ 1 − R r ​ ] Net force at B: F B ​ = f 2 + N 2 2 ​ ​ F B ​ = mg ( R r ​ ) 2 + ( 1 − R r ​ ) 2 tan 2 θ 1 ​ ​ for minimum value of μ : f ⩽ μ N 2 ​ ⇒ R mgr ​ ⩽ tan θ μ mg ​ [ 1 − R r ​ ] ⇒ μ ⩾ R / r − 1 tan θ ​ \begin{center} \end{center}

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