A uniform rod is falling without rotation on a smooth horizontal plane. Assuming the collision to be perfectly elastic, the angular velocity of the rod after striking the table is maximum when the rod makes an angle $$cos$$−$$1$$⁡$$1$$∗ with the horizontal just before striking where ∗ is not readable. Find ∗.

Question

A uniform rod is falling without rotation on a smooth horizontal plane. Assuming the collision to be perfectly elastic, the angular velocity of the rod after striking the table is maximum when the rod makes an angle $$cos$$−$$1$$⁡$$1$$∗ with the horizontal just before striking where ∗ is not readable. Find ∗.

Infinity Learn · Accepted Answer

If v be the linear velocity of rod after impact (upwards), ωbe the angular velocity of rod and J be the linear impulse at A during impact, then byImpulse $$-$$ Momentum Theorem, we have       $$J=$$Δ$$p=pf$$−pi⇒ $$J=mv$$−−$$mv0$$⇒ $$J=mv+v0$$    .........(1) Further by  Angular  Impulse − Angular  Momentum  Theorem , we have Jℓ$$2cos$$⁡θ$$=I$$ω$$=m$$ℓ$$212$$ω    ........(2)Since$$ the collision is elastic, so at the point of impact e $$=$$ $$1$$.⇒ Relative speed  of approach $$=$$ Relative speed  of separation ⇒ $$v0=v+$$ℓ$$2$$ω$$cos$$⁡θ    .......$$(3)Solving$$ equations $$(1), (2) and (3), we getω$$=6v0cos$$⁡θℓ$$1+3cos2$$⁡θFor ω to be maximum, we have    dωdθ$$=0$$⇒$$6v0$$ℓddθ$$cos$$⁡θ$$1+3cos2$$⁡θ$$=0$$⇒$$1+3cos2$$⁡θ$$($$−$$sin$$⁡θ$$)$$−$$cos$$⁡θ$$($$−$$6sin$$⁡θ$$cos$$⁡θ$$)1+3cos2$$⁡θ$$2=0$$⇒$$1+3cos2$$⁡θ$$=6cos2$$⁡θ⇒$$3cos2$$⁡θ$$=1$$⇒$$cos$$⁡θ$$=13$$⇒∗$$=3$$

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