Question Number 21604 by Tinkutara last updated on 29/Sep/17
$$\mathrm{A}\:\mathrm{particle}\:\mathrm{will}\:\mathrm{leave}\:\mathrm{a}\:\mathrm{vertical}\:\mathrm{circle}\:\mathrm{of} \\ $$$$\mathrm{radius}\:{r},\:\mathrm{when}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{at}\:\mathrm{the}\:\mathrm{lowest} \\ $$$$\mathrm{point}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\left({v}_{{L}} \right)\:\mathrm{is} \\ $$$$\left({a}\right)\:\sqrt{\mathrm{2}{gr}} \\ $$$$\left({b}\right)\:\sqrt{\mathrm{5}{gr}} \\ $$$$\left({c}\right)\:\sqrt{\mathrm{3}{gr}} \\ $$$$\left({d}\right)\:\sqrt{\mathrm{6}{gr}} \\ $$
Answered by ajfour last updated on 29/Sep/17
![at the top if it leaves the circle, then mg=m(v_H ^2 /r) or mv_H ^2 =mgr ...(i) from conservation of energy: 2mgr=(1/2)m(v_L ^2 −v_H ^2 ) ⇒ 4gr = mv_L ^2 −mgr [see (i) ] v_L = (√(5gr)) . whebever v_L < (√(5gr)) particle leaves the vertical circle. So (a) and (c) .](https://www.tinkutara.com/question/Q21618.png)
$${at}\:{the}\:{top}\:{if}\:{it}\:{leaves}\:{the}\: \\ $$$${circle},\:{then}\: \\ $$$${mg}={m}\frac{{v}_{{H}} ^{\mathrm{2}} }{{r}}\:\:\:\:{or}\:\:\:{mv}_{{H}} ^{\mathrm{2}} ={mgr}\:\:\:…\left({i}\right) \\ $$$${from}\:{conservation}\:{of}\:{energy}: \\ $$$$\mathrm{2}{mgr}=\frac{\mathrm{1}}{\mathrm{2}}{m}\left({v}_{{L}} ^{\mathrm{2}} −{v}_{{H}} ^{\mathrm{2}} \right) \\ $$$$\Rightarrow\:\:\:\:\mathrm{4}{gr}\:=\:{mv}_{{L}} ^{\mathrm{2}} −{mgr}\:\:\:\:\:\left[{see}\:\left({i}\right)\:\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:{v}_{{L}} =\:\sqrt{\mathrm{5}{gr}}\:. \\ $$$${whebever}\:{v}_{{L}} <\:\sqrt{\mathrm{5}{gr}}\:\:{particle} \\ $$$${leaves}\:{the}\:{vertical}\:{circle}. \\ $$$${So}\:\left({a}\right)\:{and}\:\left({c}\right)\:. \\ $$
Commented by Tinkutara last updated on 29/Sep/17
$$\mathrm{But}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{given}\:\mathrm{only}\:\left({c}\right)\:\mathrm{not}\:\left({a}\right). \\ $$
Commented by ajfour last updated on 29/Sep/17
$${but}\:{with}\:{v}_{{L}} =\sqrt{\mathrm{2}{gr}}\:{it}\:{swings}\:{only} \\ $$$${in}\:{the}\:{lower}\:{half}\:{vertixal}\:{circle}; \\ $$$${no}\:{chance}\:{of}\:{leaving}\:{the}\:{circle}. \\ $$$${so}\:{just}\:\left({c}\right)\:{and}\:{not}\:\left({a}\right).\:{sorry} \\ $$$${for}\:{the}\:{mistake}. \\ $$
Commented by Tinkutara last updated on 29/Sep/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by Tinkutara last updated on 29/Sep/17
$$\mathrm{Can}\:\mathrm{you}\:\mathrm{please}\:\mathrm{guide}\:\mathrm{Q}.\:\mathrm{21307}? \\ $$