A spool has the shape shown in figure. Radii of inner and outer cylinders are R and $$2$$ R respectively. Mass of the spool is $$3$$ m and its moment of inertia about the shown axis is $$2mR2$$. Light threads are tightly wrapped on both the cylindrical parts. The spool is placed on a rough surface with two masses $$m1$$ $$=$$ m and $$m2$$ $$=$$ $$2m$$ connected to the strings as shown. The string segment between spool and the pulleys $$P1$$ and $$P2$$ are horizontal. The centre of mass of the spool is at its geometrical centre. System is released from rest. What is minimum value of coefficient of friction between the spool and the table so that it does not slip?

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Infinity Learn · Accepted Answer

Let acceleration of COM of the spool be $$a0$$ towards right, and acceleration of $$m2$$ be $$a2$$ downward and that of $$m1$$ be $$a1$$ upward f $$=$$ friction, α $$=$$ acceleration of the spool $$2R$$α$$=a0$$         .......(1)R$$α$$+a0=a2$$      .....$$(2)2R$$α$$+a0=a1$$    .......$$(3)T1$$−$$mg=ma1T1$$−$$mg=ma0+2R$$α$$=2ma0$$     .....$$(4)   ∵Rα$$=a022mg$$−$$T2=2ma22mg$$−$$T2=2ma0+R$$α$$=3ma0$$    ........(5)T2+f$$−$$T1=3ma0$$                             .........$$(6)T2R$$−$$T12R$$−$$f2R=l$$α$$T2$$−$$2T1$$−$$2f=2mR$$α ∵$$I=2mR2T2$$−$$2T1$$−$$2f=ma0$$                             .......$$(7)Solving$$, $$(4), (5), (6) & (7) we get  $$f=$$−$$mg3Negative$$ sign means correct direction of friction is towards right side.∵f≤μN ∴$$mg3$$≤μ$$3mg$$ ∴$$19$$≤μ

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