Two discs A & B having small projections, welded on their circumference , are free to rotate about their fixed axes, which are parallel to each other as shown in figure. The disc B starts rotating with angular velocity ω . It’s projection hits the projection of other disc at rest after time T. The coefficient of restitution is $$12$$ . The two discs are of same radius but their moments of inertia are $$4I$$ and I respectively.The angular velocity of the disc A after collision isThe second collision between projections will take place after a minimum time of

Question

Two discs A & B having small projections, welded on their circumference , are free to rotate about their fixed axes, which are parallel to each other as shown in figure. The disc B starts rotating with angular velocity ω . It’s projection hits the projection of other disc at rest after time T. The coefficient of restitution is $$12$$ . The two discs are of same radius but their moments of inertia are $$4I$$ and I respectively.The angular velocity of the disc A after collision isThe second collision between projections will take place after a minimum time of

Infinity Learn · Accepted Answer

Let B rotates in clockwise direction initiallyJust after collision, let ω$$1$$,ω$$2$$ are their angular speeds and J is the impulse of collision. For discs⁡(1$$&$$2),$$JR=4I$$ω$$1=I$$ω$$+$$ω$$2$$⇒$$4$$ω$$1=$$ω$$+$$ω$$2$$ …….(1)e=12$$⇒$$12=R$$ω$$1$$−Rω$$2R$$ω⇒ω$$=2$$ω$$1$$−$$2$$ω$$2$$    .......$$(2)From$$ $$(1) and (2),  ω$$1=310$$ω,ω$$2=$$ω$$5$$ i.e., disc (2) reverses its direction of rotationLet left and right discs make $$N1$$, $$N2$$ number of rotations between first and $$2nd$$ collision between projections. $$N1T1=N2T2N12$$π(3$$ω$$/10)=N22$$πω$$/5$$⇒$$N1=32N2$$ Least integer values of $$N1$$ and $$N2$$ are $$N1=3$$,$$N2=2$$∴ Time of second collision $$t0=N1T1=3$$⋅$$2$$$$π$$$$3$$ω$$(10)=20T$$[∵$$T=$$π$$/$$ω]Let B rotates in clockwise direction initiallyJust after collision, let ω$$1$$,ω$$2$$ are their angular speeds and J is the impulse of collision. For discs⁡$$(1$$&$$2),$$JR=4I$$ω$$1=I$$ω$$+$$ω$$2$$⇒$$4$$ω$$1=$$ω$$+$$ω$$2$$ …….(1)e=12$$⇒$$12=R$$ω$$1$$−Rω$$2R$$ω⇒ω$$=2$$ω$$1$$−$$2$$ω$$2$$    .......$$(2)From$$ $$(1) and (2),  ω$$1=310$$ω,ω$$2=$$ω$$5$$ i.e., disc (2) reverses its direction of rotationLet left and right discs make $$N1$$, $$N2$$ number of rotations between first and $$2nd$$ collision between projections. $$N1T1=N2T2N12$$π$$(3$$ω$$/10)=N22$$πω$$/5$$⇒$$N1=32N2$$ Least integer values of $$N1$$ and $$N2$$ are $$N1=3$$,$$N2=2$$∴ Time of second collision $$t0=N1T1=3$$⋅$$2$$$$π$$$$3$$ω$$(10)=20T$$[∵$$T=$$π$$/$$ω]

General Planar motion

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The angular velocity of the disc A after collision is

The second collision between projections will take place after a minimum time of

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