A disc shaped body (having$$ a semi circular hole as shown in the $$figure) of mass $$m=10$$ kg and radius $$R=109m$$ is performing pure rolling motion on a rough horizontal surface. In the figure point O is geometrical center of the disc and at this instant the center of mass C of the disc is at same horizontal level with O. The radius of gyration of the disc about an axis passing through C and perpendicular to the plane of the disc is $$R2$$ and at the instant shown the angular velocity of the disc is ω$$=gR$$ $$rad/$$$$sec$$ in clockwise sense. g is acceleration due to gravity $$=$$ $$10$$ $$m/s2$$ . Find angular acceleration α (in$$ $$rad/$$ $$s2) of the disc at this instant.

Question

A disc shaped body (having$$ a semi circular hole as shown in the $$figure) of mass $$m=10$$ kg  and  radius   $$R=109m$$  is  performing  pure  rolling motion on  a  rough  horizontal  surface.  In the figure  point  O  is  geometrical center  of  the  disc  and  at  this  instant  the  center  of  mass  C  of  the disc is at same horizontal level with O. The radius of gyration of the disc  about  an  axis  passing    through  C  and  perpendicular  to  the  plane of the disc is  $$R2$$ and at the instant shown the angular velocity of  the  disc  is  ω$$=gR$$ $$rad/$$$$sec$$   in  clockwise  sense.  g  is  acceleration due to gravity $$=$$ $$10$$ $$m/s2$$ .  Find angular acceleration α  (in$$ $$rad/$$  $$s2) of the disc at this instant.

Infinity Learn · Accepted Answer

Considering torque of real and pseudo force in the frame of P bottom point,τ$$=mgR2+m$$ω$$2RR2$$​⇒τ$$=mgR2+mgRRR2=mgR$$​Moment of Inertia about P, $$IP=IC+m(CP)2=m(R2)2+m(R2+R24)=32mR2$$​∴   α$$=$$τ$$I=2mgR3mR2=2g3R=6$$  $$rad/s2$$

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