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Question Number 24786 by Tinkutara last updated on 26/Nov/17

A particle of mass m is moving in yz-plane  with a uniform velocity v with its  trajectory running parallel to +ve y-  axis and intersecting z-axis at z = a.  The change in its angular momentum  about the origin as it bounces elastically  from a wall at y = constant is :  (1) mvae_x ^∧   (2) 2mvae_x ^∧   (3) ymve_x ^∧   (4) 2ymve_x ^∧

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{in}\:{yz}-\mathrm{plane} \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{uniform}\:\mathrm{velocity}\:{v}\:\mathrm{with}\:\mathrm{its} \\ $$$$\mathrm{trajectory}\:\mathrm{running}\:\mathrm{parallel}\:\mathrm{to}\:+\mathrm{ve}\:{y}- \\ $$$$\mathrm{axis}\:\mathrm{and}\:\mathrm{intersecting}\:{z}-\mathrm{axis}\:\mathrm{at}\:{z}\:=\:{a}. \\ $$$$\mathrm{The}\:\mathrm{change}\:\mathrm{in}\:\mathrm{its}\:\mathrm{angular}\:\mathrm{momentum} \\ $$$$\mathrm{about}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{as}\:\mathrm{it}\:\mathrm{bounces}\:\mathrm{elastically} \\ $$$$\mathrm{from}\:\mathrm{a}\:\mathrm{wall}\:\mathrm{at}\:{y}\:=\:\mathrm{constant}\:\mathrm{is}\:: \\ $$$$\left(\mathrm{1}\right)\:{mva}\overset{\wedge} {{e}}_{{x}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2}{mva}\overset{\wedge} {{e}}_{{x}} \\ $$$$\left(\mathrm{3}\right)\:{ymv}\overset{\wedge} {{e}}_{{x}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}{ymv}\overset{\wedge} {{e}}_{{x}} \\ $$

Commented by Tinkutara last updated on 26/Nov/17

Answered by ajfour last updated on 26/Nov/17

l^� =r^� ×p^�   △l^� =r^� ×△p^�         =(yj^� +ak^� )×(−2mvj^� )        =2mvai^�   ≡ 2mvae_x ^�   option (2).

$$\bar {{l}}=\bar {{r}}×\bar {{p}} \\ $$$$\bigtriangleup\bar {{l}}=\bar {{r}}×\bigtriangleup\bar {{p}} \\ $$$$\:\:\:\:\:\:=\left({y}\hat {{j}}+{a}\hat {{k}}\right)×\left(−\mathrm{2}{mv}\hat {{j}}\right) \\ $$$$\:\:\:\:\:\:=\mathrm{2}{mva}\hat {{i}}\:\:\equiv\:\mathrm{2}{mva}\hat {{e}}_{{x}} \\ $$$${option}\:\left(\mathrm{2}\right). \\ $$

Commented by Tinkutara last updated on 26/Nov/17

But what is e_x ^∧ ?

$${But}\:{what}\:{is}\:\overset{\wedge} {{e}}_{{x}} ? \\ $$

Commented by Tinkutara last updated on 26/Nov/17

Is it same as i^∧ ?^

$${Is}\:{it}\:{same}\:{as}\:\overset{\wedge} {{i}}\overset{} {?} \\ $$

Commented by ajfour last updated on 26/Nov/17

yes.

$${yes}. \\ $$

Commented by Tinkutara last updated on 26/Nov/17

Thank you Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Sir}! \\ $$

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