Two thin circular discs of mass m and $$4m$$, having radii of a and $$2a$$ , respectively, are rigidly fixed by a massless, rigid rod of length $$l=24$$ a through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is ω . The angular momentum of the entire assembly about the point ‘O’ is L→ $$($$ see the $$figure)$$. Which of the following $$statement(s)$$ is $$(are)true$$?

Question

Two thin circular discs of mass m and $$4m$$, having radii of a  and $$2a$$  , respectively, are rigidly fixed by a massless, rigid rod of  length $$l=24$$ a through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is ω . The angular momentum of the entire assembly about the point ‘O’ is L→ $$($$ see the $$figure)$$. Which of the following $$statement(s)$$ is $$(are)true$$?

Infinity Learn · Accepted Answer

In △OAM , $$OM=l2+a2=5a$$​Circumference of a circle of radius $$OM=2$$$$π$$$$5a$$​Circumference of a smaller disc $$=2$$$$π$$aFor completing this big circle once, the smaller disc wil have to take $$2$$$$π$$$$5a2$$$$π$$$$a=5$$ rotations​Thus, the C.M. of the assembly rotates about z axis with an angular speed of ω$$5$$.The angular momentum about C.M. of the $$system=LC=IC$$ω$$=12ma2$$ω $$+$$ $$12$$×$$4m$$×$$2a2$$ω$$=17ma2$$ω$$2$$​Now, $$vc=m$$ω$$a+4m2$$ω$$a5m=9$$ω$$a5and$$ $$r1=ml+4m2l5m=9l5$$​So, L of C.M. $$=5m9$$ω$$a59l5=8124m$$ω$$a25Net$$ angular momentum about z $$axis=LZ=8124m$$ω$$a25cos$$θ − ICω$$sin$$θ​⇒$$LZ=8124m$$ω$$a25$$×$$245$$−$$17ma2$$ω$$10=3803m$$ω$$a250$$

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